# arXiv daily: Combinatorics (math.CO)

##### 1.On Ideal Secret-Sharing Schemes for $k$-homogeneous access structures

**Authors:**Younjin Kim, Jihye Kwon, Hyang-Sook Lee

**Abstract:** A $k$-uniform hypergraph is a hypergraph where each $k$-hyperedge has exactly $k$ vertices. A $k$-homogeneous access structure is represented by a $k$-uniform hypergraph $\mathcal{H}$, in which the participants correspond to the vertices of hypergraph $\mathcal{H}$. A set of vertices can reconstruct the secret value from their shares if they are connected by a $k$-hyperedge, while a set of non-adjacent vertices does not obtain any information about the secret. One parameter for measuring the efficiency of a secret sharing scheme is the information rate, defined as the ratio between the length of the secret and the maximum length of the shares given to the participants. Secret sharing schemes with an information rate equal to one are called ideal secret sharing schemes. An access structure is considered ideal if an ideal secret sharing scheme can realize it. Characterizing ideal access structures is one of the important problems in secret sharing schemes. The characterization of ideal access structures has been studied by many authors~\cite{BD, CT,JZB, FP1,FP2,DS1,TD}. In this paper, we characterize ideal $k$-homogeneous access structures using the independent sequence method. In particular, we prove that the reduced access structure of $\Gamma$ is an $(k, n)$-threshold access structure when the optimal information rate of $\Gamma$ is larger than $\frac{k-1}{k}$, where $\Gamma$ is a $k$-homogeneous access structure satisfying specific criteria.

##### 2.A survey of complex generalized weighing matrices and a construction of quantum error-correcting codes

**Authors:**Ronan Egan

**Abstract:** Some combinatorial designs, such as Hadamard matrices, have been extensively researched and are familiar to readers across the spectrum of Science and Engineering. They arise in diverse fields such as cryptography, communication theory, and quantum computing. Objects like this also lend themselves to compelling mathematics problems, such as the Hadamard conjecture. However, complex generalized weighing matrices, which generalize Hadamard matrices, have not received anything like the same level of scrutiny. Motivated by an application to the construction of quantum error-correcting codes, which we outline in the latter sections of this paper, we survey the existing literature on complex generalized weighing matrices. We discuss and extend upon the known existence conditions and constructions, and compile known existence results for small parameters. Some interesting quantum codes are constructed to demonstrate their value.

##### 3.Large Convex sets in Difference sets

**Authors:**Krishnendu Bhowmick, Ben Lund, Oliver Roche-Newton

**Abstract:** We give a construction of a convex set $A \subset \mathbb R$ with cardinality $n$ such that $A-A$ contains a convex subset with cardinality $\Omega (n^2)$. We also consider the following variant of this problem: given a convex set $A$, what is the size of the largest matching $M \subset A \times A$ such that the set \[ \{ a-b : (a,b) \in M \} \] is convex? We prove that there always exists such an $M$ with $|M| \geq \sqrt n$, and that this lower bound is best possible, up a multiplicative constant.

##### 4.Topics in Boolean Representable Simplicial Complexes

**Authors:**Stuart Margolis, John Rhodes, Pedro V. Silva

**Abstract:** We study a number of topics in the theory of Boolean Representable Simplicial Complexes (BRSC). These include various operators on BRSC. We look at shellability in higher dimensions and propose a number of new conjectures.

##### 5.Inhomogeneous order 1 iterative functional equations with applications to combinatorics

**Authors:**Lucia Di Vizio, Gwladys Fernandes, Marni Mishna

**Abstract:** We show that if a Laurent series $f\in\mathbb{C}((t))$ satisfies a particular kind of linear iterative equation, then $f$ is either an algebraic function or it is differentially transcendental over $\mathbb{C}(t)$. This condition is more precisely stated as follows: We consider $R,a,b\in \mathbb{C}(t)$ with $R(0)=0$, such that $f(R(t))=a(t)f(t)+b(t)$. If either $R'(0)=0$ or $R'(0)$ is a root of unity, then either $f$ satisfies a polynomial equation, or $f$ does not satisfy a polynomial differential equation. We illustrate how to apply this result to deduce the differential transcendence of combinatorial generating functions by considering three examples: the ordinary generating function for a family of complete trees; the Green function for excursions on the Sierpinski graph; and a series related to the enumeration of permutations avoiding the consecutive pattern 1423. The proof strategy is inspired by the Galois theory of functional equations, and relies on the property of the dynamics of $R$, Liouville-Rosenlicht's theorem and Ax' theorem.

##### 6.On Supmodular Matrices

**Authors:**Shmuel Onn

**Abstract:** We consider the problem of determining which matrices are permutable to be supmodular. We show that for small dimensions any matrix is permutable by a universal permutation or by a pair of permutations, while for higher dimensions no universal permutation exists. We raise several questions including to determine the dimensions in which every matrix is permutable.

##### 7.Random Turán and counting results for general position sets over finite fields

**Authors:**Yaobin Chen, Xizhi Liu, Jiaxi Nie, Ji Zeng

**Abstract:** Let $\alpha(\mathbb{F}_q^d,p)$ denote the maximum size of a general position set in a $p$-random subset of $\mathbb{F}_q^d$. We determine the order of magnitude of $\alpha(\mathbb{F}_q^2,p)$ up to polylogarithmic factors for all possible values of $p$, improving the previous best upper bounds obtained by Roche-Newton--Warren and Bhowmick--Roche-Newton. For $d \ge 3$ we prove upper bounds for $\alpha(\mathbb{F}_q^d,p)$ that are essentially tight within certain intervals of $p$. We establish the upper bound $2^{(1+o(1))q}$ for the number of general position sets in $\mathbb{F}_q^d$, which matches the trivial lower bound $2^{q}$ asymptotically in exponent. We also refine this counting result by proving an asymptotically tight (in exponent) upper bound for the number of general position sets with fixed size. The latter result for $d=2$ improves a result of Roche-Newton--Warren. Our proofs are grounded in the hypergraph container method, and additionally, for $d=2$ we also leverage the pseudorandomness of the point-line incidence bipartite graph of $\mathbb{F}_{q}^2$.

##### 8.On ranked and bounded Kohnert posets

**Authors:**Laura Colmenarejo, Felix Hutchins, Nicholas Mayers, Etienne Phillips

**Abstract:** In this paper, we explore combinatorial properties of the posets associated with Kohnert polynomials. In particular, we determine a sufficient condition guaranteeing when such ``Kohnert posets'' are bounded and two necessary conditions for when they are ranked. Moreover, we apply the aforementioned conditions to find complete characterizations of when Kohnert posets are bounded and when they are ranked in special cases, including those associated with Demazure characters.

##### 9.Combinatorial Proof of an Identity of Berkovich and Uncu

**Authors:**Aritram Dhar, Avi Mukhopadhyay

**Abstract:** The BG-rank BG($\pi$) of an integer partition $\pi$ is defined as $$\text{BG}(\pi) := i-j$$ where $i$ is the number of odd-indexed odd parts and $j$ is the number of even-indexed odd parts of $\pi$. In a recent work, Fu and Tang ask for a direct combinatorial proof of the following identity of Berkovich and Uncu $$B_{2N+\nu}(k,q)=q^{2k^2-k}\left[\begin{matrix}2N+\nu\\N+k\end{matrix}\right]_{q^2}$$ for any integer $k$ and non-negative integer $N$ where $\nu\in \{0,1\}$, $B_N(k,q)$ is the generating function for partitions into distinct parts less than or equal to $N$ with BG-rank equal to $k$ and $\left[\begin{matrix}a+b\\b\end{matrix}\right]_q$ is a Gaussian binomial coefficient. In this paper, we provide a combinatorial proof of Berkovich and Uncu's identity along the lines of Fu and Tang's idea.

##### 10.On faces of the Kunz cone and the numerical semigroups within them

**Authors:**Levi Borevitz, Tara Gomes, Jiajie Ma, Harper Niergarth, Christopher O'Neill, Daniel Pocklington, Rosa Stolk, Jessica Wang, Shuhang Xue

**Abstract:** A numerical semigroup is a cofinite subset of the non-negative integers that is closed under addition and contains 0. Each numerical semigroup $S$ with fixed smallest positive element $m$ corresponds to an integer point in a rational polyhedral cone $\mathcal C_m$, called the Kunz cone. Moreover, numerical semigroups corresponding to points in the same face $F \subseteq \mathcal C_m$ are known to share many properties, such as the number of minimal generators. In this work, we classify which faces of $\mathcal C_m$ contain points corresponding to numerical semigroups. Additionally, we obtain sharp bounds on the number of minimal generators of $S$ in terms of the dimension of the face of $\mathcal C_m$ containing the point corresponding to $S$.

##### 11.On an induced version of Menger's theorem

**Authors:**Kevin Hendrey, Sergey Norin, Raphael Steiner, Jérémie Turcotte

**Abstract:** We prove Menger-type results in which the obtained paths are pairwise non-adjacent, both for graphs of bounded maximum degree and, more generally, for graphs excluding a topological minor. We further show better bounds in the subcubic case, and in particular obtain a tight result for two paths using a computer-assisted proof.

##### 1.Block-and-hole graphs: Constructibility and $(3,0)$-sparsity

**Authors:**Bryan Gin-ge Chen, James Cruickshank, Derek Kitson

**Abstract:** We show that minimally 3-rigid block-and-hole graphs, with one block or one hole, are characterised as those which are constructible from $K_3$ by vertex splitting, and also, as those having associated looped face graphs which are $(3,0)$-tight. This latter property can be verified in polynomial time by a form of pebble game algorithm. We also indicate connections to the rigidity properties of polyhedral surfaces known as origami and to graph rigidity in $\ell_p^3$ for $p\not=2$.

##### 2.The complex of injective words of permutations which are not derangements is contractible

**Authors:**Assaf Libman

**Abstract:** Let $D_n \subseteq \Sigma_n$ be the set of derangements in the symmetric group. We prove that the complex of injective words generated by $\Sigma_n \setminus D_n$ is contractible. This gives a conceptual explanation to the well known fact that the complex of injective words generated by $\Sigma_n$ is homotopy equivalent to the wedge sum $\underset{|D_n|}{\bigvee} S^{n-1}$.

##### 3.An infinite family of $m$-ovoids of the hyperbolic quadrics $\mathcal{Q}^+(7,q)$

**Authors:**Francesco Pavese, Hanlin Zou

**Abstract:** An infinite family of $(q^2+q+1)$-ovoids of $\mathcal{Q}^+(7,q)$, $q\equiv 1\pmod{3}$, admitting the group $\mathrm{PGL}(3,q)$, is constructed. The main tool is the general theory of generalized hexagons.

##### 4.Cyclic 2-Spreads in $V(6,q)$ and Flag-Transitive Affine Linear Spaces

**Authors:**Cian Jameson, John Sheekey

**Abstract:** In this paper we completely classify spreads of 2-dimensional subspaces of a 6-dimensional vector space over a finite field of characteristic not two or three upon which a cyclic group acts transitively. This addresses one of the remaining open cases in the classification of flag-transitive linear spaces. We utilise the polynomial approach innovated by Pauley and Bamberg to obtain our results.

##### 5.Strong arc decompositions of split digraphs

**Authors:**Joergen Bang-Jensen, Yun Wang

**Abstract:** A {\bf strong arc decomposition} of a digraph $D=(V,A)$ is a partition of its arc set $A$ into two sets $A_1,A_2$ such that the digraph $D_i=(V,A_i)$ is strong for $i=1,2$. Bang-Jensen and Yeo (2004) conjectured that there is some $K$ such that every $K$-arc-strong digraph has a strong arc decomposition. They also proved that with one exception on 4 vertices every 2-arc-strong semicomplete digraph has a strong arc decomposition. Bang-Jensen and Huang (2010) extended this result to locally semicomplete digraphs by proving that every 2-arc-strong locally semicomplete digraph which is not the square of an even cycle has a strong arc decomposition. This implies that every 3-arc-strong locally semicomplete digraph has a strong arc decomposition. A {\bf split digraph} is a digraph whose underlying undirected graph is a split graph, meaning that its vertices can be partioned into a clique and an independent set. Equivalently, a split digraph is any digraph which can be obtained from a semicomplete digraph $D=(V,A)$ by adding a new set $V'$ of vertices and some arcs between $V'$ and $V$. In this paper we prove that every 3-arc-strong split digraph has a strong arc decomposition which can be found in polynomial time and we provide infinite classes of 2-strong split digraphs with no strong arc decomposition. We also pose a number of open problems on split digraphs.

##### 6.Three-cuts are a charm: acyclicity in 3-connected cubic graphs

**Authors:**František Kardoš, Edita Máčajová, Jean Paul Zerafa

**Abstract:** Let $G$ be a bridgeless cubic graph. In 2023, the three authors solved a conjecture (also known as the $S_4$-Conjecture) made by Mazzuoccolo in 2013: there exist two perfect matchings of $G$ such that the complement of their union is a bipartite subgraph of $G$. They actually show that given any $1^+$-factor $F$ (a spanning subgraph of $G$ such that its vertices have degree at least 1) and an arbitrary edge $e$ of $G$, there exists a perfect matching $M$ of $G$ containing $e$ such that $G\setminus (F\cup M)$ is bipartite. This is a step closer to comprehend better the Fan--Raspaud Conjecture and eventually the Berge--Fulkerson Conjecture. The $S_4$-Conjecture, now a theorem, is also the weakest assertion in a series of three conjectures made by Mazzuoccolo in 2013, with the next stronger statement being: there exist two perfect matchings of $G$ such that the complement of their union is an acyclic subgraph of $G$. Unfortunately, this conjecture is not true: Jin, Steffen, and Mazzuoccolo later showed that there exists a counterexample admitting 2-cuts. Here we show that, despite of this, every cyclically 3-edge-connected cubic graph satisfies this second conjecture.

##### 7.Limit-closed Profiles

**Authors:**Ann-Kathrin Elm, Hendrik Heine

**Abstract:** Tangle-tree theorems are an important tool in structural graph theory, and abstract separation systems are a very general setting in which tangle-tree theorems can still be formulated and proven. For infinite abstract separation systems, so far tangle-tree theorems have only been shown for special cases of separation systems, in particular when the separation system arises from a (locally finite) infinite graph. We present a tangle-tree theorem for infinite separation systems where we do not place restrictions on the separation system itself but on the tangles to be arranged in a tree.

##### 8.Turán Colourings in Off-Diagonal Ramsey Multiplicity

**Authors:**Joseph Hyde, Jae-baek Lee, Jonathan A. Noel

**Abstract:** The Ramsey multiplicity constant of a graph $H$ is the limit as $n$ tends to infinity of the minimum density of monochromatic labelled copies of $H$ in a colouring of the edges of $K_n$ with two colours. Fox and Wigderson recently identified a large family of graphs whose Ramsey multiplicity constants are attained by sequences of "Tur\'an colourings;" i.e. colourings in which one of the colour classes forms the edge set of a balanced complete multipartite graph. The graphs in their family come from taking a connected non-3-colourable graph with a critical edge and adding many pendant edges. We extend their result to an off-diagonal variant of the Ramsey multiplicity constant which involves minimizing a weighted sum of red copies of one graph and blue copies of another. We also apply the flag algebra method to investigate the minimum number of pendant edges required for Tur\'an colourings to become optimal when the underlying graphs are small cliques.

##### 9.Correlations of minimal forbidden factors of the Fibonacci word

**Authors:**Narad Rampersad, Max Wiebe

**Abstract:** If $u$ and $v$ are two words, the correlation of $u$ over $v$ is a binary word that encodes all possible overlaps between $u$ and $v$. This concept was introduced by Guibas and Odlyzko as a key element of their method for enumerating the number of words of length $n$ over a given alphabet that avoid a given set of forbidden factors. In this paper we characterize the pairwise correlations between the minimal forbidden factors of the infinite Fibonacci word.

##### 10.On the spectra of token graphs of cycles and other graphs

**Authors:**Mónica. A. Reyes, Cristina Dalfó, Miquel Àngel Fiol, Arnau Messegué

**Abstract:** The $k$-token graph $F_k(G)$ of a graph $G$ is the graph whose vertices are the $k$-subsets of vertices from $G$, two of which being adjacent whenever their symmetric difference is a pair of adjacent vertices in $G$. It is a known result that the algebraic connectivity (or second Laplacian eigenvalue) of $F_k(G)$ equals the algebraic connectivity of $G$. In this paper, we first give results that relate the algebraic connectivities of a token graph and the same graph after removing a vertex. Then, we prove the result on the algebraic connectivity of 2-token graphs for two infinite families: the odd graphs $O_r$ for all $r$, and the multipartite complete graphs $K_{n_1,n_2,\ldots,n_r}$ for all $n_1,n_2,\ldots,n_r$ In the case of cycles, we present a new method that allows us to compute the whole spectrum of $F_2(C_n)$. This method also allows us to obtain closed formulas that give asymptotically exact approximations for most of the eigenvalues of $F_2(\textit{}C_n)$.

##### 1.The $χ$-binding function of $d$-directional segment graphs

**Authors:**Lech Duraj, Ross J. Kang, Hoang La, Jonathan Narboni, Filip Pokrývka, Clément Rambaud, Amadeus Reinald

**Abstract:** Given a positive integer $d$, the class $d$-DIR is defined as all those intersection graphs formed from a finite collection of line segments in ${\mathbb R}^2$ having at most $d$ slopes. Since each slope induces an interval graph, it easily follows for every $G$ in $d$-DIR with clique number at most $\omega$ that the chromatic number $\chi(G)$ of $G$ is at most $d\omega$. We show for every even value of $\omega$ how to construct a graph in $d$-DIR that meets this bound exactly. This partially confirms a conjecture of Bhattacharya, Dvo\v{r}\'ak and Noorizadeh. Furthermore, we show that the $\chi$-binding function of $d$-DIR is $\omega \mapsto d\omega$ for $\omega$ even and $\omega \mapsto d(\omega-1)+1$ for $\omega$ odd. This refutes said conjecture of Bhattacharya, Dvo\v{r}\'ak and Noorizadeh.

##### 2.Further results on the number of cliques in graphs covered by long cycles

**Authors:**Leilei Zhang

**Abstract:** Let $\Gamma(n,k)$ be the set of $2$-connected $n$-vertex graphs containing an edge that is not on any cycle of length at least $k+1.$ Let $g_s(n,k)$ denote the maximum number of $s$-cliques in a graph in $\Gamma(n,k).$ Recently, Ji and Ye [SIAM J. Discrete Math., 37 (2023) 917-924] determined $g_s(n,k).$ They remark that it is interesting to characterize the extremal graphs. In this paper, we give such a characterization.

##### 3.Distinguishing colorings, proper colorings, and covering properties without the Axiom of Choice

**Authors:**Amitayu Banerjee, Zalán Molnár, Alexa Gopaulsingh

**Abstract:** We work with simple graphs in ZF (Zermelo--Fraenkel set theory without the Axiom of Choice (AC)) and cardinals in the absence of AC to prove that the following statements are equivalent to K\H{o}nig Lemma: (a) Any infinite locally finite connected graph G such that the minimum degree of G is greater than k, has a chromatic number for any fixed integer k greater than or equal to 2. (b) Any infinite locally finite connected graph has a chromatic index. (c) Any infinite locally finite connected graph has a distinguishing number. (d) Any infinite locally finite connected graph has a distinguishing index. Our results strengthen some results of Stawiski from a recent paper on the role of the Axiom of Choice in proper and distinguishing colorings since Stawiski worked with cardinals in the presence of AC. We also formulate new conditions for the existence of irreducible proper coloring, minimal edge cover, maximal matching, and minimal dominating set in connected bipartite graphs and locally finite connected graphs, which are either equivalent to AC or K\H{o}nig Lemma. Moreover, we show that if the Axiom of Choice for families of 2 element sets holds, then the Shelah--Soifer graph has a minimal dominating set.

##### 4.Two involutions on binary trees and generalizations

**Authors:**Yang Li, Zhicong Lin, Tongyuan Zhao

**Abstract:** This paper investigates two involutions on binary trees. One is the mirror symmetry of binary trees which combined with the classical bijection $\varphi$ between binary trees and plane trees answers an open problem posed by Bai and Chen. This involution can be generalized to weakly increasing trees, which admits to merge two recent equidistributions found by Bai--Chen and Chen--Fu, respectively. The other one is constructed to answer a bijective problem on di-sk trees asked by Fu--Lin--Wang and can be generalized naturally to rooted labeled trees. This second involution combined with $\varphi$ leads to a new statistic on plane trees whose distribution gives the Catalan's triangle. Moreover, a quadruple equidistribution on plane trees involving this new statistic is proved via a recursive bijection.

##### 5.Lower bounds on the homology of Vietoris-Rips complexes of hypercube graphs

**Authors:**Henry Adams, Žiga Virk

**Abstract:** We provide novel lower bounds on the Betti numbers of Vietoris-Rips complexes of hypercube graphs of all dimensions, and at all scales. In more detail, let $Q_n$ be the vertex set of $2^n$ vertices in the $n$-dimensional hypercube graph, equipped with the shortest path metric. Let $VR(Q_n;r)$ be its Vietoris--Rips complex at scale parameter $r \ge 0$, which has $Q_n$ as its vertex set, and all subsets of diameter at most $r$ as its simplices. For integers $r<r'$ the inclusion $VR(Q_n;r)\hookrightarrow VR(Q_n;r')$ is nullhomotopic, meaning no persistent homology bars have length longer than one, and we therefore focus attention on the individual spaces $VR(Q_n;r)$. We provide lower bounds on the ranks of homology groups of $VR(Q_n;r)$. For example, using cross-polytopal generators, we prove that the rank of $H_{2^r-1}(VR(Q_n;r))$ is at least $2^{n-(r+1)}\binom{n}{r+1}$. We also prove a version of \emph{homology propagation}: if $q\ge 1$ and if $p$ is the smallest integer for which $rank H_q(VR(Q_p;r))\neq 0$, then $rank H_q(VR(Q_n;r)) \ge \sum_{i=p}^n 2^{i-p} \binom{i-1}{p-1} \cdot rank H_q(VR(Q_p;r))$ for all $n \ge p$. When $r\le 3$, this result and variants thereof provide tight lower bounds on the rank of $H_q(VR(Q_n;r))$ for all $n$, and for each $r \ge 4$ we produce novel lower bounds on the ranks of homology groups. Furthermore, we show that for each $r\ge 2$, the homology groups of $VR(Q_n;r)$ for $n \ge 2r+1$ contain propagated homology not induced by the initial cross-polytopal generators.

##### 6.The zero forcing span of a graph

**Authors:**Bonnie Jacob

**Abstract:** In zero forcing, the focus is typically on finding the minimum cardinality of any zero forcing set in the graph; however, the number of cardinalities between $0$ and the number of vertices in the graph for which there are both zero forcing sets and sets that fail to be zero forcing sets is not well known. In this paper, we introduce the zero forcing span of a graph, which is the number of distinct cardinalities for which there are sets that are zero forcing sets and sets that are not. We introduce the span within the context of standard zero forcing and skew zero forcing as well as for standard zero forcing on directed graphs. We characterize graphs with high span and low span of each type, and also investigate graphs with special zero forcing polynomials.

##### 7.Lipschitz harmonic functions on vertex-transitive graphs

**Authors:**Gideon Amir, Guy Blachar, Maria Gerasimova, Gady Kozma

**Abstract:** We prove that every locally finite vertex-transitive graph $G$ admits a non-constant Lipschitz harmonic function.

##### 8.On the evolution of random integer compositions

**Authors:**David Bevan, Dan Threlfall

**Abstract:** We explore how the asymptotic structure of a random $n$-term weak integer composition of $m$ evolves, as $m$ increases from zero. The primary focus is on establishing thresholds for the appearance and disappearance of substructures. These include the longest and shortest runs of zero terms or of nonzero terms, longest increasing runs, longest runs of equal terms, largest squares (runs of $k$ terms each equal to $k$), as well as a wide variety of other patterns. Of particular note is the dichotomy between the appearance and disappearance of exact consecutive patterns, with smaller patterns appearing before larger ones, whereas longer patterns disappear before shorter ones.

##### 9.Eccentric graph of trees and their Cartesian products

**Authors:**Anita Arora, Rajiv Mishra

**Abstract:** Let $G$ be an undirected simple connected graph. We say a vertex $u$ is eccentric to a vertex $v$ in $G$ if $d(u,v)=\max\{d(v,w): w\in V(G)\}$. The eccentric graph, $E(G)$ of $G$ is a graph defined on the same vertex set as of $G$ and two vertices are adjacent if one is eccentric to the other. We find the structure and the girth of the eccentric graph of trees and see that the girth of the eccentric graph of a tree can either be zero, three, or four. Further, we study the structure of the eccentric graph of the Cartesian product of graphs and prove that the girth of the eccentric graph of the Cartesian product of trees can only be zero, three, four or six. Furthermore, we provide a comprehensive classification when the eccentric girth assumes these values. We also give the structure of the eccentric graph of the grid graphs and the Cartesian product of cycles. Finally, we determine the conditions under which the eccentricity matrix of the Cartesian product of trees becomes invertible.

##### 10.Diagonal operators, $q$-Whittaker functions and rook theory

**Authors:**Samrith Ram, Michael J. Schlosser

**Abstract:** We discuss the problem posed by Bender, Coley, Robbins and Rumsey of enumerating the number of subspaces which have a given profile with respect to a linear operator over the finite field $\mathbb{F}_q$. We solve this problem in the case where the operator is diagonalizable. The solution leads us to a new class of polynomials $b_{\mu\nu}(q)$ indexed by pairs of integer partitions. These polynomials have several interesting specializations and can be expressed as positive sums over semistandard tableaux. We present a new correspondence between set partitions and semistandard tableaux. A close analysis of this correspondence reveals the existence of several new set partition statistics which generate the polynomials $b_{\mu\nu}(q)$; each such statistic arises from a Mahonian statistic on multiset permutations. The polynomials $b_{\mu\nu}(q)$ are also given a description in terms of coefficients in the monomial expansion of $q$-Whittaker symmetric functions which are specializations of Macdonald polynomials. We express the Touchard--Riordan generating polynomial for chord diagrams by number of crossings in terms of $q$-Whittaker functions. We also introduce a class of $q$-Stirling numbers defined in terms of the polynomials $b_{\mu\nu}(q)$ and present connections with $q$-rook theory in the spirit of Garsia and Remmel.

##### 1.l-connectivity, l-edge-connectivity and spectral radius of graphs

**Authors:**Dandan Fan, Xiaofeng Gu, Huiqiu Lin

**Abstract:** Let G be a connected graph. The toughness of G is defined as t(G)=min{\frac{|S|}{c(G-S)}}, in which the minimum is taken over all proper subsets S\subset V(G) such that c(G-S)\geq 2 where c(G-S) denotes the number of components of G-S. Confirming a conjecture of Brouwer, Gu [SIAM J. Discrete Math. 35 (2021) 948--952] proved a tight lower bound on toughness of regular graphs in terms of the second largest absolute eigenvalue. Fan, Lin and Lu [European J. Combin. 110 (2023) 103701] then studied the toughness of simple graphs from the spectral radius perspective. While the toughness is an important concept in graph theory, it is also very interesting to study |S| for which c(G-S)\geq l for a given integer l\geq 2. This leads to the concept of the l-connectivity, which is defined to be the minimum number of vertices of G whose removal produces a disconnected graph with at least l components or a graph with fewer than l vertices. Gu [European J. Combin. 92 (2021) 103255] discovered a lower bound on the l-connectivity of regular graphs via the second largest absolute eigenvalue. As a counterpart, we discover the connection between the l-connectivity of simple graphs and the spectral radius. We also study similar problems for digraphs and an edge version.

##### 2.On Ahn-Hendrey-Kim-Oum question for twin-width of graphs with 6 vertices

**Authors:**Kajal Das

**Abstract:** Twin-width is a recently introduced graph parameter for finite graphs. It is an open problem to determine whether there is an $n$-vertex graph having twin-width at least $n/2$ (due to J. Ahn, K. Hendrey, D. Kim and S. Oum). In an earlier paper, the author showed that such a graph with less than equal to 5 vertices does not exist. In this article, we show that such a graph with 6 vertices does not exist. More precisely, we prove that each graph with 6 vertices has twin-width less than equal to 2.

##### 3.Commutator nilpotency for somewhere-to-below shuffles

**Authors:**Darij Grinberg

**Abstract:** Given a positive integer $n$, we consider the group algebra of the symmetric group $S_{n}$. In this algebra, we define $n$ elements $t_{1},t_{2},\ldots,t_{n}$ by the formula \[ t_{\ell}:=\operatorname*{cyc}\nolimits_{\ell}+\operatorname*{cyc}\nolimits_{\ell,\ell+1}+\operatorname*{cyc}\nolimits_{\ell,\ell+1,\ell+2}+\cdots+\operatorname*{cyc}\nolimits_{\ell,\ell+1,\ldots,n}, \] where $\operatorname*{cyc}\nolimits_{\ell,\ell+1,\ldots,k}$ denotes the cycle that sends $\ell\mapsto\ell+1\mapsto\ell+2\mapsto\cdots\mapsto k\mapsto\ell$. These $n$ elements are called the *somewhere-to-below shuffles* due to an interpretation as card-shuffling operators. In this paper, we show that their commutators $\left[ t_{i},t_{j}\right] =t_{i}t_{j}-t_{j}t_{i}$ are nilpotent, and specifically that \[ \left[ t_{i},t_{j}\right] ^{\left\lceil \left( n-j\right) /2\right\rceil +1}=0\ \ \ \ \ \ \ \ \ \ \text{for any }i,j\in\left\{ 1,2,\ldots,n\right\} \] and \[ \left[ t_{i},t_{j}\right] ^{j-i+1}=0\ \ \ \ \ \ \ \ \ \ \text{for any }1\leq i\leq j\leq n. \] We discuss some further identities and open questions.

##### 4.The number of $1$-nearly independent vertex subsets

**Authors:**Eric Ould Dadah Andriantiana, Zekhaya B. Shozi

**Abstract:** Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. A subset $I$ of $V(G)$ is an independent vertex subset if no two vertices in $I$ are adjacent in $G$. We study the number, $\sigma_1(G)$, of all subsets of $v(G)$ that contain exactly one pair of adjacent vertices. We call those subsets 1-nearly independent vertex subsets. Recursive formulas of $\sigma_1$ are provided, as well as some cases of explicit formulas. We prove a tight lower (resp. upper) bound on $\sigma_1$ for graphs of order $n$. We deduce as a corollary that the star $K_{1,n-1}$ (the tree with degree sequence $(n-1,1,\dots,1)$) is the $n$-vertex tree with smallest $\sigma_1$, while it is well known that $K_{1,n-1}$ is the $n$-vertex tree with largest number of independent subsets.

##### 5.Universality for graphs of bounded degeneracy

**Authors:**Peter Allen, Julia Böttcher, Anita Liebenau

**Abstract:** Given a family $\mathcal{H}$ of graphs, a graph $G$ is called $\mathcal{H}$-universal if $G$ contains every graph of $\mathcal{H}$ as a subgraph. Following the extensive research on universal graphs of small size for bounded-degree graphs, Alon asked what is the minimum number of edges that a graph must have to be universal for the class of all $n$-vertex graphs that are $D$-degenerate. In this paper, we answer this question up to a factor that is polylogarithmic in $n.$

##### 6.The Rational Number Game

**Authors:**Nathan Bowler, Florian Gut

**Abstract:** We investigate a game played between two players, Maker and Breaker, on a countably infinite complete graph where the vertices are the rational numbers. The players alternately claim unclaimed edges. It is Maker's goal to have after countably many turns a complete infinite graph contained in her coloured edges where the vertex set of the subgraph is order-isomorphic to the rationals. It is Breaker's goal to prevent Maker from achieving this. We prove that there is a winning strategy for Maker in this game. We also prove that there is a winning strategy for Breaker in the game where Maker must additionally make the vertex set of her complete graph dense in the rational numbers.

##### 7.Distribution of colours in rainbow H-free colourings

**Authors:**Zhuo Wu, Jun Yan

**Abstract:** An edge colouring of $K_n$ with $k$ colours is a Gallai $k$-colouring if it does not contain any rainbow triangle. Gy\'arf\'as, P\'alv\"olgyi, Patk\'os and Wales proved that there exists a number $g(k)$ such that $n\geq g(k)$ if and only if for any colour distribution sequence $(e_1,\cdots,e_k)$ with $\sum_{i=1}^ke_i=\binom{n}{2}$, there exist a Gallai $k$-colouring of $K_n$ with $e_i$ edges having colour $i$. They also showed that $\Omega(k)=g(k)=O(k^2)$ and posed the problem of determining the exact order of magnitude of $g(k)$. Feffer, Fu and Yan improved both bounds significantly by proving $\Omega(k^{1.5}/\log k)=g(k)=O(k^{1.5})$. We resolve this problem by showing $g(k)=\Theta(k^{1.5}/(\log k)^{0.5})$. Moreover, we generalise these definitions by considering rainbow $H$-free colourings of $K_n$ for any general graph $H$, and the natural corresponding quantity $g(H,k)$. We prove that $g(H,k)$ is finite for every $k$ if and only if $H$ is not a forest, and determine the order of $g(H,k)$ when $H$ contains a subgraph with minimum degree at least 3.

##### 8.Flag-Shaped Blockers of 123-Avoiding Permutation Matrices

**Authors:**Megan Bennett, Lei Cao

**Abstract:** A blocker of $123$-avoiding permutation matrices refers to the set of zeros contained within an $n\times n$ $123$-forcing matrix. Recently, Brualdi and Cao provided a characterization of all minimal blockers, which are blockers with a cardinality of $n$. Building upon their work, a new type of blocker, flag-shaped blockers, which can be seen as a generalization of the $L$-shaped blockers defined by Brualdi and Cao, are introduced. It is demonstrated that all flag-shaped blockers are minimum blockers. The possible cardinalities of flag-shaped blockers are also determined, and the dimensions of subpolytopes that are defined by flag-shaped blockers are examined.

##### 1.Almost partitioning every $2$-edge-coloured complete $k$-graph into $k$ monochromatic tight cycles

**Authors:**Allan Lo, Vincent Pfenninger

**Abstract:** A $k$-uniform tight cycle is a $k$-graph with a cyclic order of its vertices such that every $k$ consecutive vertices from an edge. We show that for $k\geq 3$, every red-blue edge-coloured complete $k$-graph on $n$ vertices contains $k$ vertex-disjoint monochromatic tight cycles that together cover $n - o(n)$ vertices.

##### 2.Equivariant theory for codes and lattices I

**Authors:**Himadri Shekhar Chakraborty, Tsuyoshi Miezaki

**Abstract:** In this paper, we present a generalization of Hayden's theorem [7, Theorem 4.2] for $G$-codes over finite Frobenius rings. A lattice theoretical form of this generalization is also given. Moreover, Astumi's MacWilliams identity [1, Theorem 1] is generalized in several ways for different weight enumerators of $G$-codes over finite Frobenius rings. Furthermore, we provide the Jacobi analogue of Astumi's MacWilliams identity for $G$-codes over finite Frobenius rings. Finally, we study the relation between $G$-codes and its corresponding $G$-lattices.

##### 3.Star Colouring of Bounded Degree Graphs and Regular Graphs

**Authors:**Shalu M. A., Cyriac Antony

**Abstract:** A $k$-star colouring of a graph $G$ is a function $f:V(G)\to\{0,1,\dots,k-1\}$ such that $f(u)\neq f(v)$ for every edge $uv$ of $G$, and every bicoloured connected subgraph of $G$ is a star. The star chromatic number of $G$, $\chi_s(G)$, is the least integer $k$ such that $G$ is $k$-star colourable. We prove that $\chi_s(G)\geq \lceil (d+4)/2\rceil$ for every $d$-regular graph $G$ with $d\geq 3$. We reveal the structure and properties of even-degree regular graphs $G$ that attain this lower bound. The structure of such graphs $G$ is linked with a certain type of Eulerian orientations of $G$. Moreover, this structure can be expressed in the LC-VSP framework of Telle and Proskurowski (SIDMA, 1997), and hence can be tested by an FPT algorithm with the parameter either treewidth, cliquewidth, or rankwidth. We prove that for $p\geq 2$, a $2p$-regular graph $G$ is $(p+2)$-star colourable only if $n:=|V(G)|$ is divisible by $(p+1)(p+2)$. For each $p\geq 2$ and $n$ divisible by $(p+1)(p+2)$, we construct a $2p$-regular Hamiltonian graph on $n$ vertices which is $(p+2)$-star colourable. The problem $k$-STAR COLOURABILITY takes a graph $G$ as input and asks whether $G$ is $k$-star colourable. We prove that 3-STAR COLOURABILITY is NP-complete for planar bipartite graphs of maximum degree three and arbitrarily large girth. Besides, it is coNP-hard to test whether a bipartite graph of maximum degree eight has a unique 3-star colouring up to colour swaps. For $k\geq 3$, $k$-STAR COLOURABILITY of bipartite graphs of maximum degree $k$ is NP-complete, and does not even admit a $2^{o(n)}$-time algorithm unless ETH fails.

##### 4.Boundary rigidity of 3D CAT(0) cube complexes

**Authors:**John Haslegrave, Alex Scott, Youri Tamitegama, Jane Tan

**Abstract:** The boundary rigidity problem is a classical question from Riemannian geometry: if $(M, g)$ is a Riemannian manifold with smooth boundary, is the geometry of $M$ determined up to isometry by the metric $d_g$ induced on the boundary $\partial M$? In this paper, we consider a discrete version of this problem: can we determine the combinatorial type of a finite cube complex from its boundary distances? As in the continuous case, reconstruction is not possible in general, but one expects a positive answer under suitable contractibility and non-positive curvature conditions. Indeed, in two dimensions Haslegrave gave a positive answer to this question when the complex is a finite quadrangulation of the disc with no internal vertices of degree less than $4$. We prove a $3$-dimensional generalisation of this result: the combinatorial type of a finite CAT(0) cube complex with an embedding in $\mathbb{R}^3$ can be reconstructed from its boundary distances. Additionally, we prove a direct strengthening of Haslegrave's result: the combinatorial type of any finite 2-dimensional CAT(0) cube complex can be reconstructed from its boundary distances.

##### 5.Induced subgraphs and tree decompositions XI. Local structure in even-hole-free graph of large treewidth

**Authors:**Bogdan Alecu, Maria Chudnovsky, Sepehr Hajebi, Sophie Spirkl

**Abstract:** Sintiari and Trotignon showed that for every $h\geq 1$, there are (even-hole, $K_4$)-free graphs of arbitrarily large treewidth in which every $h$-vertex induced subgraph is chordal. We prove the converse: given a graph $H$, every (even-hole, $K_4$)-free graph of large enough treewidth contains an induced subgraph isomorphic to $H$, if and only if $H$ is chordal (and $K_4$-free). As an immediate corollary, the above result settles a conjecture of Sintiari and Trotignon, asserting that every (even-hole, $K_4$)-free graph of sufficiently large treewidth contains an induced subgraph isomorphic to the graph obtained from the two-edge path by adding a universal vertex (also known as the "diamond"). We further prove yet another extension of their conjecture with "$K_4$" replaced by an arbitrary complete graph and the "two-edge path" replaced by an arbitrary forest. This turns out to characterize forests: given a graph $F$, for every $t\geq 1$, every (even-hole, $K_t$)-free graph of sufficiently large treewidth contains an induced subgraph isomorphic to the graph obtained from $F$ by adding a universal vertex, if and only if $F$ is a forest.

##### 6.Density of $3$-critical signed graphs

**Authors:**Laurent Beaudou, Penny Haxell, Kathryn Nurse, Sagnik Sen, Zhouningxin Wang

**Abstract:** We say that a signed graph is $k$-critical if it is not $k$-colorable but every one of its proper subgraphs is $k$-colorable. Using the definition of colorability due to Naserasr, Wang, and Zhu that extends the notion of circular colorability, we prove that every $3$-critical signed graph on $n$ vertices has at least $\frac{3n-1}{2}$ edges, and that this bound is asymptotically tight. It follows that every signed planar or projective-planar graph of girth at least $6$ is (circular) $3$-colorable, and for the projective-planar case, this girth condition is best possible. To prove our main result, we reformulate it in terms of the existence of a homomorphism to the signed graph $C_{3}^*$, which is the positive triangle augmented with a negative loop on each vertex.

##### 7.Essentially tight bounds for rainbow cycles in proper edge-colourings

**Authors:**Noga Alon, Matija Bucić, Lisa Sauermann, Dmitrii Zakharov, Or Zamir

**Abstract:** An edge-coloured graph is said to be rainbow if no colour appears more than once. Extremal problems involving rainbow objects have been a focus of much research over the last decade as they capture the essence of a number of interesting problems in a variety of areas. A particularly intensively studied question due to Keevash, Mubayi, Sudakov and Verstra\"ete from 2007 asks for the maximum possible average degree of a properly edge-coloured graph on $n$ vertices without a rainbow cycle. Improving upon a series of earlier bounds, Tomon proved an upper bound of $(\log n)^{2+o(1)}$ for this question. Very recently, Janzer-Sudakov and Kim-Lee-Liu-Tran independently removed the $o(1)$ term in Tomon's bound, showing a bound of $O(\log^2 n)$. We prove an upper bound of $(\log n)^{1+o(1)}$ for this maximum possible average degree when there is no rainbow cycle. Our result is tight up to the $o(1)$ term, and so it essentially resolves this question. In addition, we observe a connection between this problem and several questions in additive number theory, allowing us to extend existing results on these questions for abelian groups to the case of non-abelian groups.

##### 1.On the third ABC index of trees and unicyclic graphs

**Authors:**Rui Song

**Abstract:** Let $G=(V,E)$ be a simple connected graph with vertex set $V(G)$ and edge set $E(G)$. The third atom-bond connectivity index, $ABC_3$ index, of $G$ is defined as $ABC_3(G)=\sum\limits_{uv\in E(G)}\sqrt{\frac{e(u)+e(v)-2}{e(u)e(v)}}$, where eccentricity $e(u)$ is the largest distance between $u$ and any other vertex of $G$, namely $e(u)=\max\{d(u,v)|v\in V(G)\}$. This work determines the maximal $ABC_3$ index of unicyclic graphs with any given girth and trees with any given diameter, and characterizes the corresponding graphs.

##### 2.Boxicity and Interval-Orders: Petersen and the Complements of Line Graphs

**Authors:**Marco Caoduro, András Sebő

**Abstract:** The boxicity of a graph is the smallest dimension $d$ allowing a representation of it as the intersection graph of a set of $d$-dimensional axis-parallel boxes. We present a simple general approach to determining the boxicity of a graph based on studying its ``interval-order subgraphs''. The power of the method is first tested on the boxicity of some popular graphs that have resisted previous attempts: the boxicity of the Petersen graph is $3$, and more generally, that of the Kneser-graphs $K(n,2)$ is $n-2$ if $n\ge 5$, confirming a conjecture of Caoduro and Lichev [Discrete Mathematics, Vol. 346, 5, 2023]. Since every line graph is an induced subgraph of the complement of $K(n,2)$, the developed tools show furthermore that line graphs have only a polynomial number of edge-maximal interval-order subgraphs. This opens the way to polynomial-time algorithms for problems that are in general $\mathcal{NP}$-hard: for the existence and optimization of interval-order subgraphs of line-graphs, or of interval-completions of their complement.

##### 3.The $K^4$-Game

**Authors:**Nathan Bowler, Florian Gut

**Abstract:** We investigate a two player game called the $K^4$-building game: two players alternately claim edges of an infinite complete graph. Each player's aim is to claim all six edges on some vertex set of size four for themself. The first player to accomplish this goal is declared the winner of the game. We present a winning strategy which guarantees a win for the first player.

##### 4.A combinatorial view on star moments of regular directed graphs and trees

**Authors:**Benjamin Dadoun, Patrick Oliveira Santos

**Abstract:** We investigate the method of moments for $d$-regular digraphs and the limiting $d$-regular directed tree $T_d$ as the number of vertices tends to infinity, in the same spirit as McKay (Linear Algebra Appl., 1981) for the undirected setting. In particular, we provide a combinatorial derivation of the formula for the star moments (from a root vertex $o\in T_d$) $$M_d(w)\qquad:=\sum_{\substack{v_0,v_1\ldots,v_{k-1},v_k\in T_d\\v_0=v_k=o}} A^{w_1}(v_0,v_1)A^{w_2}(v_1,v_2) \cdots A^{w_k}(v_{k-1},v_k)$$ with $A$ the adjacency matrix of $T_d$, where $w:=w_1\cdots w_k$ is any word on the alphabet $\{1,{*}\}$ and $A^*$ is the adjoint matrix of $A$. Our analysis highlights a connection between the non-zero summands of $M_d(w)$ and the non-crossing partitions of $\{1,\ldots,k\}$ which are in some sense compatible with $w$.

##### 5.LDP polygons and the number 12 revisited

**Authors:**Ulrike Bücking, Christian Haase, Karin Schaller, Jan-Hendrik de Wiljes

**Abstract:** We give a combinatorial proof of a lattice point identity involving a lattice polygon and its dual, generalizing the formula $area(\Delta) + area(\Delta^*) = 6$ for reflexive $\Delta$. The identity is equivalent to the stringy Libgober-Wood identity for toric log del Pezzo surfaces.

##### 6.Some remarks on off-diagonal Ramsey numbers for vector spaces over $\mathbb{F}_{2}$

**Authors:**Zach Hunter, Cosmin Pohoata

**Abstract:** For every positive integer $d$, we show that there must exist an absolute constant $c > 0$ such that the following holds: for any integer $n \geq cd^{7}$ and any red-blue coloring of the one-dimensional subspaces of $\mathbb{F}_{2}^{n}$, there must exist either a $d$-dimensional subspace for which all of its one-dimensional subspaces get colored red or a $2$-dimensional subspace for which all of its one-dimensional subspaces get colored blue. This answers recent questions of Nelson and Nomoto, and confirms that for any even plane binary matroid $N$, the class of $N$-free, claw-free binary matroids is polynomially $\chi$-bounded. Our argument will proceed via a reduction to a well-studied additive combinatorics problem, originally posed by Green: given a set $A \subset \mathbb{F}_{2}^{n}$ with density $\alpha \in [0,1]$, what is the largest subspace that we can find in $A+A$? Our main contribution to the story is a new result for this problem in the regime where $1/\alpha$ is large with respect to $n$, which utilizes ideas from the recent breakthrough paper of Kelley and Meka on sets of integers without three-term arithmetic progressions.

##### 1.A lattice on Dyck paths close to the Tamari lattice

**Authors:**Jean-Luc Baril, Sergey Kirgizov, Mehdi Naima

**Abstract:** We introduce a new poset structure on Dyck paths where the covering relation is a particular case of the relation inducing the Tamari lattice. We prove that the transitive closure of this relation endows Dyck paths with a lattice structure. We provide a trivariate generating function counting the number of Dyck paths with respect to the semilength, the numbers of outgoing and incoming edges in the Hasse diagram. We deduce the numbers of coverings, meet and join irreducible elements. As a byproduct, we present a new involution on Dyck paths that transports the bistatistic of the numbers of outgoing and incoming edges into its reverse. Finally, we give a generating function for the number of intervals, and we compare this number with the number of intervals in the Tamari lattice.

##### 2.Bumpless pipe dreams meet Puzzles

**Authors:**Neil J. Y. Fan, Peter L. Guo, Rui Xiong

**Abstract:** Knutson and Zinn-Justin recently found a puzzle rule for the expansion of the product $\mathfrak{G}_{u}(x,t)\cdot \mathfrak{G}_{v}(x,t)$ of two double Grothendieck polynomials indexed by permutations with separated descents. We establish its triple Schubert calculus version in the sense of Knutson and Tao, namely, a formula for expanding $\mathfrak{G}_{u}(x,y)\cdot \mathfrak{G}_{v}(x,t)$ in different secondary variables. Our rule is formulated in terms of pipe puzzles, incorporating both the structures of bumpless pipe dreams and classical puzzles. As direct applications, we recover the separated-descent puzzle formula by Knutson and Zinn-Justin (by setting $y=t$) and the bumpless pipe dream model of double Grothendieck polynomials by Weigandt (by setting $v=\operatorname{id}$ and $x=t$). Moreover, we utilize the formula to partially confirm a positivity conjecture of Kirillov about applying a skew operator to a Schubert polynomial.

##### 3.Hardinian Arrays

**Authors:**Robert Dougherty-Bliss, Manuel Kauers

**Abstract:** In 2014, R.H. Hardin contributed a family of sequences about king-moves on an array to the On-Line Encyclopedia of Integer Sequences (OEIS). The sequences were recently noticed in an automated search of the OEIS by Kauers and Koutschan, who conjectured a recurrence for one of them. We prove their conjecture as well as some older conjectures stated in the OEIS entries. We also have some new conjectures for the asymptotics of Hardin's sequences.

##### 4.Small weight codewords of projective geometric codes II

**Authors:**Sam Adriaensen, Lins Denaux

**Abstract:** The $p$-ary linear code $\mathcal C_{k}(n,q)$ is defined as the row space of the incidence matrix $A$ of $k$-spaces and points of $\text{PG}(n,q)$. It is known that if $q$ is square, a codeword of weight $q^k\sqrt{q}+\mathcal O \left( q^{k-1} \right) $ exists that cannot be written as a linear combination of at most $\sqrt{q}$ rows of $A$. Over the past few decades, researchers have put a lot of effort towards proving that any codeword of smaller weight does meet this property. We show that if $ q \geqslant 32 $ is a composite prime power, every codeword of $\mathcal C_k(n,q)$ up to weight $\mathcal O \left( {q^k\sqrt{q}} \right) $ is a linear combination of at most $\sqrt{q}$ rows of $A$. We also generalise this result to the codes $\mathcal C_{j,k}(n,q) $, which are defined as the $p$-ary row span of the incidence matrix of $k$-spaces and $j$-spaces, $j < k$.

##### 5.Tensor products of multimatroids and a Brylawski-type formula for the transition polynomial

**Authors:**Iain Moffatt, Steven Noble, Maya Thompson

**Abstract:** Brylawski's tensor product formula expresses the Tutte polynomial of the tensor product of two graphs in terms of Tutte polynomials arising from the tensor factors. We are concerned with extensions of Brylawski's tensor product formula to the Bollobas-Riordan and transition polynomials of graphs embedded in surfaces. We give a tensor product formula for the multimatroid transition polynomial and show that Brylawski's formula and its topological analogues arise as specialisations of this more general result.

##### 1.The seating couple problem in even case

**Authors:**M. Meszka, A. Pasotti, M. A. Pellegrini

**Abstract:** In this paper we consider the seating couple problem with an even number of seats, which, using graph theory terminology, can be stated as follows. Given a positive even integer $v=2n$ and a list $L$ containing $n$ positive integers not exceeding $n$, is it always possible to find a perfect matching of $K_v$ whose list of edge-lengths is $L$? Up to now a (non-constructive) solution is known only when all the edge-lengths are coprime with $v$. In this paper we firstly present some necessary conditions for the existence of a solution. Then, we give a complete constructive solution when the list consists of one or two distinct elements, and when the list consists of consecutive integers $1,2,\ldots,x$, each one appearing with the same multiplicity. Finally, we propose a conjecture and some open problems.

##### 2.On size Ramsey numbers for a pair of cycles

**Authors:**Małgorzata Bednarska-Bzdęga, Tomasz Łuczak

**Abstract:** We show that there exists an absolute constant $A$ such that the size Ramsey number of a pair of cycles $(C_n$, $C_{2d})$, where $4\le 2d\le n$, is bounded from above by $An$. We also study the restricted size Ramsey number for such a pair.

##### 3.Sketches, moves and partitions: counting regions of deformations of reflection arrangements

**Authors:**Priyavrat Deshpande, Krishna Menon

**Abstract:** The collection of reflecting hyperplanes of a finite Coxeter group is called a reflection arrangement and it appears in many subareas of combinatorics and representation theory. We focus on the problem of counting regions of reflection arrangements and their deformations. Inspired by the recent work of Bernardi, we show that the notion of moves and sketches can be used to provide a uniform and explicit bijection between regions of (the Catalan deformation of) a reflection arrangement and certain non-nesting partitions. We then use the exponential formula to describe a statistic on these partitions such that distribution is given by the coefficients of the characteristic polynomial. Finally, we consider a sub-arrangement of type C arrangement called the threshold arrangement and its Catalan and Shi deformations.

##### 4.Distance-regular graphs with classical parameters that support a uniform structure: case $q \ge 2$

**Authors:**Blas Fernández, Roghayeh Maleki, Štefko Miklavič, Giusy Monzillo

**Abstract:** Let $\Gamma=(X,\mathcal{R})$ denote a finite, simple, connected, and undirected non-bipartite graph with vertex set $X$ and edge set $\mathcal{R}$. Fix a vertex $x \in X$, and define $\mathcal{R}_f = \mathcal{R} \setminus \{yz \mid \partial(x,y) = \partial(x,z)\}$, where $\partial$ denotes the path-length distance in $\Gamma$. Observe that the graph $\Gamma_f=(X,\mathcal{R}_f)$ is bipartite. We say that $\Gamma$ supports a uniform structure with respect to $x$ whenever $\Gamma_f$ has a uniform structure with respect to $x$ in the sense of Miklavi\v{c} and Terwilliger \cite{MikTer}. Assume that $\Gamma$ is a distance-regular graph with classical parameters $(D,q,\alpha,\beta)$ and diameter $D\geq 4$. Recall that $q$ is an integer such that $q \not \in \{-1,0\}$. The purpose of this paper is to study when $\Gamma$ supports a uniform structure with respect to $x$. We studied the case $q \le 1$ in \cite{FMMM}, and so in this paper we assume $q \geq 2$. Let $T=T(x)$ denote the Terwilliger algebra of $\Gamma$ with respect to $x$. Under an additional assumption that every irreducible $T$-module with endpoint $1$ is thin, we show that if $\Gamma$ supports a uniform structure with respect to $x$, then either $\alpha = 0$ or $\alpha=q$, $\beta=q^2(q^D-1)/(q-1)$, and $D \equiv 0 \pmod{6}$.

##### 5.Forbidden subgraphs and complete partitions

**Authors:**John Byrne, Michael Tait, Craig Timmons

**Abstract:** A graph is called an $(r,k)$-graph if its vertex set can be partitioned into $r$ parts of size at most $k$ with at least one edge between any two parts. Let $f(r,H)$ be the minimum $k$ for which there exists an $H$-free $(r,k)$-graph. In this paper we build on the work of Axenovich and Martin, obtaining improved bounds on this function when $H$ is a complete bipartite graph, even cycle, or tree. Some of these bounds are best possible up to a constant factor and confirm a conjecture of Axenovich and Martin in several cases. We also generalize this extremal problem to uniform hypergraphs and prove some initial results in that setting.

##### 6.Some new results on Minuscule polynomial of type A

**Authors:**Ming-Jian Ding, Jiang Zeng

**Abstract:** Bourn and Erickson (arXiv:2307.02652) recently studied a polynomial $N_n(x)$ connecting the earth mover's distance to minuscule lattices of Type A, coined Minuscule polynomial of type A in this paper. They proved that this polynomial is palindromic and unimodal, and conjectured its real-rootedness as well as a remarkable formula when $x=1$. In this paper, we shall confirm these conjectures and further prove that the coefficients are asymptotically normal and the coefficient matrix of $N_n(x)$ is totally positive.

##### 7.Towards the Overfull Conjecture

**Authors:**Songling Shan

**Abstract:** Let $G$ be a simple graph with maximum degree denoted as $\Delta(G)$. An overfull subgraph $H$ of $G$ is a subgraph satisfying the condition $|E(H)| > \Delta(G)\lfloor \frac{1}{2}|V(H)| \rfloor$. In 1986, Chetwynd and Hilton proposed the Overfull Conjecture, stating that a graph $G$ with maximum degree $\Delta(G)> \frac{1}{3}|V(G)|$ has chromatic index equal to $\Delta(G)$ if and only if it does not contain any overfull subgraph. The Overfull Conjecture has many implications. For example, it implies a polynomial-time algorithm for determining the chromatic index of graphs $G$ with $\Delta(G) > \frac{1}{3}|V(G)|$, and implies several longstanding conjectures in the area of graph edge colorings. In this paper, we make the first improvement towards the conjecture when not imposing a minimum degree condition on the graph: for any $0<\varepsilon \le \frac{1}{22}$, there exists a positive integer $n_0$ such that if $G$ is a graph on $n\ge n_0$ vertices with $\Delta(G) \ge (1-\varepsilon)n$, then the Overfull Conjecture holds for $G$. The previous best result in this direction, due to Chetwynd and Hilton from 1989, asserts the conjecture for graphs $G$ with $\Delta(G) \ge |V(G)|-3$.

##### 8.Limited packings: related vertex partitions and duality issues

**Authors:**Azam Sadat Ahmadi, Nasrin Soltankhah, Babak Samadi

**Abstract:** A $k$-limited packing partition ($k$LP partition) of a graph $G$ is a partition of $V(G)$ into $k$-limited packing sets. We consider the $k$LP partitions with minimum cardinality (with emphasis on $k=2$). The minimum cardinality is called $k$LP partition number of $G$ and denoted by $\chi_{\times k}(G)$. This problem is the dual problem of $k$-tuple domatic partitioning as well as a generalization of the well-studied $2$-distance coloring problem in graphs. We give the exact value of $\chi_{\times2}$ for trees and bound it for general graphs. A section of this paper is devoted to the dual of this problem, where we give a solution to an open problem posed in $1998$. We also revisit the total limited packing number in this paper and prove that the problem of computing this parameter is NP-hard even for some special families of graphs. We give some inequalities concerning this parameter and discuss the difference between $2$TLP number and $2$LP number with emphasis on trees.

##### 9.Stress-linked pairs of vertices and the generic stress matroid

**Authors:**Dániel Garamvölgyi

**Abstract:** Given a graph $G$ and a mapping $p : V(G) \rightarrow \mathbb{R}^d$, we say that the pair $(G,p)$ is a ($d$-dimensional) realization of $G$. Two realizations $(G,p)$ and $(G,q)$ are equivalent if each of the point pairs corresponding to the edges of $G$ have the same distance under the embeddings $p$ and $q$. A pair of vertices $\{u,v\}$ is globally linked in $G$ in $\mathbb{R}^d$ if for every generic realization $(G,p)$ and every equivalent realization $(G,q)$, $(G+uv,p)$ and $(G+uv,q)$ are also equivalent. In this paper we introduce the notion of $d$-stress-linked vertex pairs. Roughly speaking, a pair of vertices $\{u,v\}$ is $d$-stress-linked in $G$ if the edge $uv$ is generically stressed in $G+uv$ and for every generic $d$-dimensional realization $(G,p)$, every configuration $q$ that satisfies all of the equilibrium stresses of $(G,p)$ also satisfies the equilibrium stresses of $(G+uv,p)$. Among other results, we show that $d$-stress-linked vertex pairs are globally linked in $\mathbb{R}^d$, and we give a combinatorial characterization of $2$-stress-linked vertex pairs that matches the conjecture of Jackson et al. about the characterization of globally linked pairs in $\mathbb{R}^2$. As a key tool, we introduce and study the "algebraic dual" of the $d$-dimensional generic rigidity matroid of a graph, which we call the $d$-dimensional generic stress matroid of the graph. We believe that our results about this matroid, which describes the global behaviour of equilibrium stresses of generic realizations of $G$, may be of independent interest. We use our results to give positive answers to conjectures of Jord\'an, Connelly, and Grasegger et al.

##### 10.Chromatic number of spacetime

**Authors:**James Davies

**Abstract:** We prove that every finite colouring of $\mathbb{Q}^3 \subset \mathbb{R}^3$ contains a monochromatic pair of points $(x,y,z),(x',y',z')$ with $(x-x')^2 + (y-y')^2 - (z-z')^2 = 1$.

##### 1.On Card guessing games: limit law for no feedback one-time riffle shuffle

**Authors:**Markus Kuba, Alois Panholzer

**Abstract:** We consider the following card guessing game with no feedback. An ordered deck of n cards labeled 1 up to n is riffle-shuffled exactly one time. Then, the goal of the game is to maximize the number of correct guesses of the cards. One after another a single card is drawn from the top, the guesser makes a guess without seeing the card and gets no response if the guess was correct or not. Building upon and improving earlier results, we provide a limit law for the number of correct guesses and also show convergence of the integer moments.

##### 2.$λ$-quiddit{é}s sur des produits directs d'anneaux

**Authors:**Flavien Mabilat
LMR

**Abstract:** The aim of this article is to continue the study of the notion of $\lambda$-quiddity over a ring, which appeared during the study of Coxeter's friezes. For this, we will focus here on situations where the ring used can be seen as a direct product of unitary commutative rings. In particular, we will consider the cases of direct products of rings containing at least two rings of characteristic 0 and we will also consider some products of the type $\mathbb{Z}/n\mathbb{Z} \times \mathbb{Z}/m\mathbb{Z}$.

##### 3.Weak$^*$ degeneracy and weak$^*$ $k$-truncated-degree-degenerate graphs

**Authors:**Huan Zhou, Jialu Zhu, Xuding Zhu

**Abstract:** This paper introduces the concept of weak$^*$ degeneracy of a graph that shares many nice properties of degeneracy. We prove that for any $f: V(G) \to \mathbb{N}$, if $G$ is weak$^*$ $f$-degenerate, then $G$ is DP-$f$-paintable and $f$-AT. Hence the weak$^*$ degeneracy of $G$ is an upper bound for many colouring parameters, including the online DP-chromatic number and the Alon-Tarsi number. Let $k$ be a positive integer and let $f(v)=\min\{d_G(v), k\}$ for each vertex $v$ of $G$. If $G$ is weak$^*$ $f$-degenerate (respectively, $f$-choosable), then we say $G$ is weak$^*$ $k$-truncated-degree--degenerate (respectively, $k$-truncated-degree-choosable). Richtor asked whether every 3-connected non-complete planar graph is $6$-truncated-degree-choosable. We construct a 3-connected non-complete planar graph which is not $7$-truncated-degree-choosable, so the answer to Richtor's question is negative even if 6 is replaced by 7. Then we prove that every 3-connected non-complete planar graph is weak$^*$ $16$-truncated-degree-degenerate (and hence $16$-truncated-degree-choosable). For an arbitrary proper minor closed family ${\mathcal G}$ of graphs, let $s$ be the minimum integer such that $K_{s,t} \notin \mathcal{G}$ for some $t$. We prove that there is a constant $k$ such that every $s$-connected non-complete graph $G \in {\mathcal G}$ is weak$^*$ $k$-truncated-degree-degenerate. In particular, for any surface $\Sigma$, there is a constant $k$ such that every 3-connected non-complete graph embeddable on $\Sigma$ is weak$^*$ $k$-truncated-degree-degenerate.

##### 4.Forbidden patterns of graphs 12-representable by pattern-avoiding words

**Authors:**Asahi Takaoka

**Abstract:** A graph $G = (\{1, 2, \ldots, n\}, E)$ is $12$-representable if there is a word $w$ over $\{1, 2, \ldots, n\}$ such that two vertices $i$ and $j$ with $i < j$ are adjacent if and only if every $j$ occurs before every $i$ in $w$. These graphs have been shown to be equivalent to the complements of simple-triangle graphs. This equivalence provides a characterization in terms of forbidden patterns in vertex orderings as well as a polynomial-time recognition algorithm. The class of $12$-representable graphs was introduced by Jones et al. (2015) as a variant of word-representable graphs. A general research direction for word-representable graphs suggested by Kitaev and Lozin (2015) is to study graphs representable by some specific types of words. For instance, Gao, Kitaev, and Zhang (2017) and Mandelshtam (2019) investigated word-representable graphs represented by pattern-avoiding words. Following this research direction, this paper studies $12$-representable graphs represented by words that avoid a pattern $p$. Such graphs are trivial when $p$ is of length $2$. When $p = 111$, $121$, $231$, and $321$, the classes of such graphs are equivalent to well-known classes, such as trivially perfect graphs and bipartite permutation graphs. For the cases where $p = 123$, $132$, and $211$, this paper provides forbidden pattern characterizations.

##### 5.Decomposing random regular graphs into stars

**Authors:**Michelle Delcourt, Catherine Greenhill, Mikhail Isaev, Bernard Lidický, Luke Postle

**Abstract:** We study $k$-star decompositions, that is, partitions of the edge set into disjoint stars with $k$ edges, in the uniformly random $d$-regular graph model $\mathcal{G}_{n,d}$. We prove an existence result for such decompositions for all $d,k$ such that $d/2 < k \leq d/2 + \max\{1,\frac{1}{6}\log d\}$. More generally, we give a sufficient existence condition that can be checked numerically for any given values of $d$ and $k$. Complementary negative results are obtained using the independence ratio of random regular graphs. Our results establish an existence threshold for $k$-star decompositions in $\mathcal{G}_{n,d}$ for all $d\leq 100$ and $k > d/2$, and strongly suggest the a.a.s. existence of such decompositions is equivalent to the a.a.s. existence of independent sets of size $(2k-d)n/(2k)$, subject to the necessary divisibility conditions on the number of vertices. For smaller values of $k$, the connection between $k$-star decompositions and $\beta$-orientations allows us to apply results of Thomassen (2012) and Lov\'asz, Thomassen, Wu and Zhang (2013). We prove that random $d$-regular graphs satisfy their assumptions with high probability, thus establishing a.a.s. existence of $k$-star decompositions (i) when $2k^2+k\leq d$, and (ii) when $k$ is odd and $k < d/2$.

##### 6.Asymptotics of Some Plancherel Averages via Polynomiality Results

**Authors:**Werner Schachinger

**Abstract:** Consider Young diagrams of $n$ boxes distributed according to the Plancherel measure. So those diagrams could be the output of the RSK algorithm, when applied to random permutations of the set $\{1,\ldots,n\}$. Here we are interested in asymptotics, as $n\to \infty$, of expectations of certain functions of random Young diagrams, such as the number of bumping steps of the RSK algorithm that leads to that diagram, the side length of its Durfee square, or the logarithm of its probability. We can express these functions in terms of hook lengths or contents of the boxes of the diagram, which opens the door for application of known polynomiality results for Plancherel averages. We thus obtain representations of expectations as binomial convolutions, that can be further analyzed with the help of Rice's integral or Poisson generating functions. Among our results is a very explicit expression for the constant appearing in the almost equipartition property of the Plancherel measure.

##### 7.Behavior of the Minimum Degree Throughout the $d$-process

**Authors:**Jakob Hofstad

**Abstract:** The $d$-process generates a graph at random by starting with an empty graph with $n$ vertices, then adding edges one at a time uniformly at random among all pairs of vertices which have degrees at most $d-1$ and are not mutually joined. We show that, in the evolution of a random graph with $n$ vertices under the $d$-process, with high probability, for each $j \in \{0,1,\dots,d-2\}$, the minimum degree jumps from $j$ to $j+1$ when there are $\Theta(\ln(n)^{d-j-1})$ steps left. This answers a question of Ruci\'nski. More specifically, we show that, when the last vertex of degree $j$ disappears, the number of steps left divided by $\ln(n)^{d-j-1}$ converges in distribution to the exponential random variable of mean $\frac{j!}{2(d-1)!}$; furthermore, these $d-1$ distributions are independent.

##### 8.The extremal number of cycles with all diagonals

**Authors:**Domagoj Bradač, Abhishek Methuku, Benny Sudakov

**Abstract:** In 1975, Erd\H{o}s asked the following natural question: What is the maximum number of edges that an $n$-vertex graph can have without containing a cycle with all diagonals? Erd\H{o}s observed that the upper bound $O(n^{5/3})$ holds since the complete bipartite graph $K_{3,3}$ can be viewed as a cycle of length six with all diagonals. In this paper, we resolve this old problem. We prove that there exists a constant $C$ such that every $n$-vertex with $Cn^{3/2}$ edges contains a cycle with all diagonals. Since any cycle with all diagonals contains cycles of length four, this bound is best possible using well-known constructions of graphs without a four-cycle based on finite geometry. Among other ideas, our proof involves a novel lemma about finding an `almost-spanning' robust expander which might be of independent interest.

##### 1.A representation of a set of maps as a ribbon bipartite graph

**Authors:**Yury Kochetkov

**Abstract:** In this purely experimental work we try to represent the set of plane maps with 3 vertices and 3 faces as a bipartite ribbon graph. In particular, this construction allows one to estimate the genus of the initial set.

##### 2.2-Coupon Coloring of Cubic Graphs Containing 3-Cycle or 4-Cycle

**Authors:**S. Akbari, M. Azimian, A. Fazli Khani, B. Samimi, E. Zahiri

**Abstract:** Let $G$ be a graph. A total dominating set in a graph $G$ is a set $S$ of vertices of $G$ such that every vertex in $G$ is adjacent to a vertex in $S$. Recently, the following question was proposed: "Is it true that every connected cubic graph containing a $3$-cycle has two vertex disjoint total dominating sets?" In this paper, we give a negative answer to this question. Moreover, we prove that if we replace $3$-cycle with $4$-cycle the answer is affirmative. This implies every connected cubic graph containing a diamond (the complete graph of order $4$ minus one edge) as a subgraph can be partitioned into two total dominating sets, a result that was proved in 2017.

##### 3.A note on transverse sets and bilinear varieties

**Authors:**Luka Milićević

**Abstract:** Let $G$ and $H$ be finite-dimensional vector spaces over $\mathbb{F}_p$. A subset $A \subseteq G \times H$ is said to be transverse if all of its rows $\{x \in G \colon (x,y) \in A\}$, $y \in H$, are subspaces of $G$ and all of its columns $\{y \in H \colon (x,y) \in A\}$, $x \in G$, are subspaces of $H$. As a corollary of a bilinear version of Bogolyubov argument, Gowers and the author proved that dense transverse sets contain bilinear varieties of bounded codimension. In this paper, we provide a direct combinatorial proof of this fact. In particular, we improve the bounds and evade the use of Fourier analysis and Freiman's theorem and its variants.

##### 4.Distance Labeling for Families of Cycles

**Authors:**Arseny M. Shur, Mikhail Rubinchik

**Abstract:** For an arbitrary finite family of graphs, the distance labeling problem asks to assign labels to all nodes of every graph in the family in a way that allows one to recover the distance between any two nodes of any graph from their labels. The main goal is to minimize the number of unique labels used. We study this problem for the families $\mathcal{C}_n$ consisting of cycles of all lengths between 3 and $n$. We observe that the exact solution for directed cycles is straightforward and focus on the undirected case. We design a labeling scheme requiring $\frac{n\sqrt{n}}{\sqrt{6}}+O(n)$ labels, which is almost twice less than is required by the earlier known scheme. Using the computer search, we find an optimal labeling for each $n\le 17$, showing that our scheme gives the results that are very close to the optimum.

##### 5.On the chromatic number of some ($P_3\cup P_2$)-free graphs

**Authors:**Rui Li, Jinfeng Li, Di Wu

**Abstract:** A hereditary class $\cal G$ of graphs is {\em $\chi$-bounded} if there is a {\em $\chi$-binding function}, say $f$, such that $\chi(G)\le f(\omega(G))$ for every $G\in\cal G$, where $\chi(G)(\omega(G))$ denotes the chromatic (clique) number of $G$. It is known that for every $(P_3\cup P_2)$-free graph $G$, $\chi(G)\le \frac{1}{6}\omega(G)(\omega(G)+1)(\omega(G)+2)$ \cite{BA18}, and the class of $(2K_2, 3K_1)$-free graphs does not admit a linear $\chi$-binding function\cite{BBS19}. In this paper, we prove that (\romannumeral 1) $\chi(G)\le2\omega(G)$ if $G$ is ($P_3\cup P_2$, kite)-free, (\romannumeral 2) $\chi(G)\le\omega^2(G)$ if $G$ is ($P_3\cup P_2$, hammer)-free, (\romannumeral 3) $\chi(G)\le\frac{3\omega^2(G)+\omega(G)}{2}$ if $G$ is ($P_3\cup P_2, C_5$)-free. Furthermore, we also discuss $\chi$-binding functions for $(P_3\cup P_2, K_4)$-free graphs.

##### 6.Asymptotics of Reciprocal Supernorm Partition Statistics

**Authors:**Jeffrey C. Lagarias, Chenyang Sun

**Abstract:** We consider two multiplicative statistics on the set of integer partitions: the norm of a partition, which is the product of its parts, and the supernorm of a partition, which is the product of the prime numbers $p_i$ indexed by its parts $i$. We introduce and study new statistics that are sums of reciprocals of supernorms on three statistical ensembles of partitions, labelled by their size $|\lambda|=n$, their perimeter equaling $n$, and their largest part equaling $n$. We show that the cumulative statistics of the reciprocal supernorm for each of the three ensembles are asymptotic to $e^{\gamma} \log n$ as $n \to \infty$.

##### 7.Calligraphs and sphere realizations

**Authors:**Matteo Gallet, Georg Grasegger, Niels Lubbes, Josef Schicho

**Abstract:** We introduce a recursive procedure for computing the number of realizations of a minimally rigid graph on the sphere up to rotations. We accomplish this by combining two ingredients. The first is a framework that allows us to think of such realizations as of elements of a moduli space of stable rational curves with marked points. The second is the idea of splitting a minimally rigid graph into two subgraphs, called calligraphs, that admit one degree of freedom and that share only a single edge and a further vertex. This idea has been recently employed for realizations of graphs in the plane up to isometries. The key result is that we can associate to a calligraph a triple of natural numbers with a special property: whenever a minimally rigid graph is split into two calligraphs, the number of realizations of the former equals the product of the two triples of the latter, where this product is specified by a fixed quadratic form. These triples and quadratic form codify the fact that we express realizations as intersections of two curves on the blowup of a sphere along two pairs of complex conjugate points.

##### 8.The power of many colours

**Authors:**Noga Alon, Matija Bucić, Micha Cristoph, Michael Krivelevich

**Abstract:** A classical problem, due to Gerencs\'er and Gy\'arf\'as from 1967, asks how large a monochromatic connected component can we guarantee in any $r$-edge colouring of $K_n$? We consider how big a connected component can we guarantee in any $r$-edge colouring of $K_n$ if we allow ourselves to use up to $s$ colours. This is actually an instance of a more general question of Bollob\'as from about 20 years ago which asks for a $k$-connected subgraph in the same setting. We complete the picture in terms of the approximate behaviour of the answer by determining it up to a logarithmic term, provided $n$ is large enough. We obtain more precise results for certain regimes which solve a problem of Liu, Morris and Prince from 2007, as well as disprove a conjecture they pose in a strong form. We also consider a generalisation in a similar direction of a question first considered by Erd\H{o}s and R\'enyi in 1956, who considered given $n$ and $m$, what is the smallest number of $m$-cliques which can cover all edges of $K_n$? This problem is essentially equivalent to the question of what is the minimum number of vertices that are certain to be incident to at least one edge of some colour in any $r$-edge colouring of $K_n$. We consider what happens if we allow ourselves to use up to $s$ colours. We obtain a more complete understanding of the answer to this question for large $n$, in particular determining it up to a constant factor for all $1\le s \le r$, as well as obtaining much more precise results for various ranges including the correct asymptotics for essentially the whole range.

##### 9.On the existence of small strictly Neumaier graphs

**Authors:**Aida Abiad, Maarten De Boeck, Sjanne Zeijlemaker

**Abstract:** A Neumaier graph is a non-complete edge-regular graph containing a regular clique. In this work, we prove several results on the existence of small strictly Neumaier graphs. In particular, we present a theoretical proof of the uniqueness of the smallest Neumaier graphs with parameters $(16,9,4;2,4)$, we establish the existence of $(25,12,5;2,5)$, and we disprove the existence of Neumaier graphs with parameters $(25,16,9;3,5)$, $(28,18,11;4,7)$, $(33,24,17;6,9)$, $(35,22,12;3,5)$ and $(55,30,18;3,5)$. Our proofs use combinatorial techniques and a novel application of integer programming methods.

##### 1.Distance-regular Cayley graphs over $\mathbb{Z}_{p^s}\oplus\mathbb{Z}_{p}$

**Authors:**Xiongfeng Zhan, Lu Lu, Xueyi Huang

**Abstract:** In [Distrance-regular Cayley graphs on dihedral groups, J. Combin. Theory Ser B 97 (2007) 14--33], Miklavi\v{c} and Poto\v{c}nik proposed the problem of characterizing distance-regular Cayley graphs, which can be viewed as an extension of the problem of identifying strongly regular Cayley graphs, or equivalently, regular partial difference sets. In this paper, all distance-regular Cayley graphs over $\mathbb{Z}_{p^s}\oplus\mathbb{Z}_{p}$ with $p$ being an odd prime are determined. It is shown that every such graph is isomorphic to a complete graph, a complete multipartite graph, or the line graph of a transversal design $TD(r,p)$ with $2\leq r\leq p-1$.

##### 2.Polyhedral combinatorics of bisectors

**Authors:**Aryaman Jal, Katharina Jochemko

**Abstract:** For any polyhedral norm, the bisector of two points is a polyhedral complex. We study combinatorial aspects of this complex. We investigate the sensitivity of the presence of labelled maximal cells in the bisector relative to the position of the two points. We thereby extend work of Criado, Joswig and Santos (2022) who showed that for the tropical distance function the presence of maximal cells is encoded by a polyhedral fan, the bisection fan. We initiate the study of bisection cones and bisection fans with respect to arbitrary polyhedral norms. In particular, we show that the bisection fan always exists for polyhedral norms in two dimensions. Furthermore, we determine the bisection fan of the $\ell_{1}$-norm and the $\ell_{\infty}$-norm as well as the discrete Wasserstein distance in arbitrary dimensions. Intricate combinatorial structures, such as the resonance arrangement, make their appearance. We apply our results to obtain bounds on the combinatorial complexity of the bisectors.

##### 3.Mutual visibility in hypercube-like graphs

**Authors:**Serafino Cicerone, Alessia Di Fonso, Gabriele Di Stefano, Alfredo Navarra, Francesco Piselli

**Abstract:** Let $G$ be a graph and $X\subseteq V(G)$. Then, vertices $x$ and $y$ of $G$ are $X$-visible if there exists a shortest $u,v$-path where no internal vertices belong to $X$. The set $X$ is a mutual-visibility set of $G$ if every two vertices of $X$ are $X$-visible, while $X$ is a total mutual-visibility set if any two vertices from $V(G)$ are $X$-visible. The cardinality of a largest mutual-visibility set (resp. total mutual-visibility set) is the mutual-visibility number (resp. total mutual-visibility number) $\mu(G)$ (resp. $\mu_t(G)$) of $G$. It is known that computing $\mu(G)$ is an NP-complete problem, as well as $\mu_t(G)$. In this paper, we study the (total) mutual-visibility in hypercube-like networks (namely, hypercubes, cube-connected cycles, and butterflies). Concerning computing $\mu(G)$, we provide approximation algorithms for both hypercubes and cube-connected cycles, while we give an exact formula for butterflies. Concerning computing $\mu_t(G)$ (in the literature, already studied in hypercubes), we provide exact formulae for both cube-connected cycles and butterflies.

##### 4.A Decomposition of Cylindric Partitions and Cylindric Partitions into Distinct Parts

**Authors:**Kağan Kurşungöz, Halime Ömrüuzun Seyrek

**Abstract:** We show that cylindric partitions are in one-to-one correspondence with a pair which has an ordinary partition and a colored partition into distinct parts. Then, we show the general form of the generating function for cylindric partitions into distinct parts and give some examples. We prove part of a conjecture by Corteel, Dousse, and Uncu. With due computational support; the other part of Corteel, Dousse, and Uncu's conjecture, which also appeared in Warnaar's work, is extended. The approaches and proofs are elementary and combinatorial.

##### 5.On linear preservers of permanental rank

**Authors:**Alexander Guterman, Igor Spiridonov

**Abstract:** Let ${\rm Mat}_n(\mathbb{F})$ denote the set of square $n\times n$ matrices over a field $\mathbb{F}$ of characteristic different from two. The permanental rank ${\rm prk}\,(A)$ of a matrix $A \in{\rm Mat}_{n}(\mathbb{F})$ is the size of the maximal square submatrix in $A$ with nonzero permanent. By $\Lambda^{k}$ and $\Lambda^{\leq k}$ we denote the subsets of matrices $A \in {\rm Mat}_{n}(\mathbb{F})$ with ${\rm prk}\,(A) = k$ and ${\rm prk}\,(A) \leq k$, respectively. In this paper for each $1 \leq k \leq n-1$ we obtain a complete characterization of linear maps $T: {\rm Mat}_{n}(\mathbb{F}) \to {\rm Mat}_{n}(\mathbb{F})$ satisfying $T(\Lambda^{\leq k}) = \Lambda^{\leq k}$ or bijective linear maps satisfying $T(\Lambda^{\leq k}) \subseteq \Lambda^{\leq k}$. Moreover, we show that if $\mathbb{F}$ is an infinite field, then $\Lambda^{k}$ is Zariski dense in $\Lambda^{\leq k}$ and apply this to describe such bijective linear maps satisfying $T(\Lambda^{k}) \subseteq \Lambda^{k}$.

##### 6.Expected Number of Dice Rolls Until an Increasing Run of Three

**Authors:**Daniel Chen

**Abstract:** A closed form is found for the expected number of rolls of a fair n-sided die until three consecutive increasing values are seen. The answer is rational, and the greatest common divisor of the numerator and denominator is given in terms of n. As n goes to infinity, the probability generating function is found for the limiting case, which is also the exponential generating function for permutations ending in a double rise and without other double rises. Thus exact values are found for the limiting expectation and variance, which are approximately 7.92437 and 27.98133 respectively.

##### 7.Twin-width of graphs with tree-structured decompositions

**Authors:**Irene Heinrich, Simon Raßmann

**Abstract:** The twin-width of a graph measures its distance to co-graphs and generalizes classical width concepts such as tree-width or rank-width. Since its introduction in 2020 (Bonnet et. al. 2020), a mass of new results has appeared relating twin width to group theory, model theory, combinatorial optimization, and structural graph theory. We take a detailed look at the interplay between the twin-width of a graph and the twin-width of its components under tree-structured decompositions: We prove that the twin-width of a graph is at most twice its strong tree-width, contrasting nicely with the result of (Bonnet and D\'epr\'es 2022), which states that twin-width can be exponential in tree-width. Further, we employ the fundamental concept from structural graph theory of decomposing a graph into highly connected components, in order to obtain an optimal linear bound on the twin-width of a graph given the widths of its biconnected components. For triconnected components we obtain a linear upper bound if we add red edges to the components indicating the splits which led to the components. Extending this approach to quasi-4-connectivity, we obtain a quadratic upper bound. Finally, we investigate how the adhesion of a tree decomposition influences the twin-width of the decomposed graph.

##### 1.Constructing and sampling partite, $3$-uniform hypergraphs with given degree sequence

**Authors:**Andras Hubai, Tamas Robert Mezei, Ferenc Beres, Andras Benczur, Istvan Miklos

**Abstract:** Partite, $3$-uniform hypergraphs are $3$-uniform hypergraphs in which each hyperedge contains exactly one point from each of the $3$ disjoint vertex classes. We consider the degree sequence problem of partite, $3$-uniform hypergraphs, that is, to decide if such a hypergraph with prescribed degree sequences exists. We prove that this decision problem is NP-complete in general, and give a polynomial running time algorithm for third almost-regular degree sequences, that is, when each degree in one of the vertex classes is $k$ or $k-1$ for some fixed $k$, and there is no restriction for the other two vertex classes. We also consider the sampling problem, that is, to uniformly sample partite, $3$-uniform hypergraphs with prescribed degree sequences. We propose a Parallel Tempering method, where the hypothetical energy of the hypergraphs measures the deviation from the prescribed degree sequence. The method has been implemented and tested on synthetic and real data. It can also be applied for $\chi^2$ testing of contingency tables. We have shown that this hypergraph-based $\chi^2$ test is more sensitive than the standard $\chi^2$ test. The extra sensitivity is especially advantageous on small data sets, where the proposed Parallel Tempering method shows promising performance.

##### 2.A lower bound on the multicolor size-Ramsey numbers of paths in hypergraphs

**Authors:**Deepak Bal, Louis DeBiasio, Allan Lo

**Abstract:** The $r$-color size-Ramsey number of a $k$-uniform hypergraph~$H$, denoted by $\hat{R}_r(H)$, is the minimum number of edges in a $k$-uniform hypergraph $G$ such that for every $r$-coloring of the edges of $G$ there exists a monochromatic copy of $H$. When $H$ is a graph path, it is known that $\Omega(r^2n)=\hat{R}_r(P_n)=O((r^2\log r)n)$ with the best bounds essentially due to Krivelevich. Letzter, Pokrovskiy, and Yepremyan~\cite{LPY} recently proved that, for the $k$-uniform tight path $P_{n}^{(k)}$, $\hat{R}_r(P_{n}^{(k)})=O_{r,k}(n)$. We consider the problem of giving a lower bound on $\hat{R}_r(P_{n}^{(k)})$ (for fixed $k$ and growing $r$). We show that $\hat{R}_r(P_n^{(k)})=\Omega_k(r^kn)$. In the case $k=3$, we give a more precise estimate which in particular improves the best known lower bound for $2$ colors due to Winter; i.e. we show $\hat{R}(P^{(3)}_{n})\geq \frac{28}{9}n-30$. All of our results above generalize to $\ell$-overlapping $k$-uniform paths $P_{n}^{(k, \ell)}$. In general we have $\hat{R}_r ( P_{n}^{(k, \ell)} ) = \Omega_k( r^{\left\lfloor \frac{k}{k-\ell} \right\rfloor } n)$, and when $1\leq \ell\leq \frac{k}{2}$ we have $\hat{R}_r(P_{n}^{(k, \ell)})=\Omega_k(r^{2}n)$.

##### 3.The critical group of a combinatorial map

**Authors:**Criel Merino, Iain Moffatt, Steven Noble

**Abstract:** Motivated by the appearance of embeddings in the theory of chip firing and the critical group of a graph, we introduce a version of the critical group (or sandpile group) for combinatorial maps, that is, for graphs embedded in orientable surfaces. We provide several definitions of our critical group, by approaching it through analogues of the cycle-cocycle matrix, the Laplacian matrix, and as the group of critical states of a chip firing game (or sandpile model) on the edges of a map. Our group can be regarded as a perturbation of the classical critical group of its underlying graph by topological information, and it agrees with the classical critical group in the plane case. Its cardinality is equal to the number of spanning quasi-trees in a connected map, just as the cardinality of the classical critical group is equal to the number of spanning trees of a connected graph. Our approach exploits the properties of principally unimodular matrices and the methods of delta-matroid theory.

##### 4.Simonovits's theorem in random graphs

**Authors:**Ilay Hoshen, Wojciech Samotij

**Abstract:** Let $H$ be a graph with $\chi(H) = r+1$. Simonovits's theorem states that, if $H$ is edge-critical, the unique largest $H$-free subgraph of $K_n$ is its largest $r$-partite subgraph, provided that $n$ is sufficiently large. We show that the same holds with $K_n$ replaced by the binomial random graph $G_{n,p}$ whenever $H$ is also strictly $2$-balanced and $p \ge (\theta_H+o(1)) n^{-\frac{1}{m_2(H)}} (\log n)^{\frac{1}{e_H-1}}$ for some explicit constant $\theta_H$, which we believe to be optimal. This (partially) resolves a conjecture of DeMarco and Kahn.

##### 5.Super FiboCatalan Numbers and Generalized FiboCatalan Numbers

**Authors:**Kendra Killpatrick

**Abstract:** Catalan observed in 1874 that the numbers $S(m,n) = \frac{(2m)! (2n)!}{m! n! (m+n)!}$, now called the super Catalan numbers, are integers but there is still no known combinatorial interpretation for them in general, although interpretations have been given for the case $m=2$ and for $S(m, m+s)$ for $0 \leq s \leq 3$. In this paper, we define the super FiboCatalan numbers $S(m,n)_F = \frac{F_{2m}! F_{2n}!}{F_m! F_n! F_{m+n}!}$ and prove they are integers for $m=1$ and $m=2$. In addition, we prove that $S(m, m+s)_F$ is an integer for $0 \leq s \leq 4$.

##### 6.On nonrepetitive colorings of paths and cycles

**Authors:**Fábio Botler, Wanderson Lomenha, João Pedro de Souza

**Abstract:** We say that a sequence $a_1 \cdots a_{2t}$ of integers is repetitive if $a_i = a_{i+t}$ for every $i\in\{1,\ldots,t\}$. A walk in a graph $G$ is a sequence $v_1 \cdots v_r$ of vertices of $G$ in which $v_iv_{i+1}\in E(G)$ for every $i\in\{1,\ldots,r-1\}$. Given a $k$-coloring $c\colon V(G)\to\{1,\ldots,k\}$ of $V(G)$, we say that $c$ is walk-nonrepetitive (resp. stroll-nonrepetitive) if for every $t\in\mathbb{N}$ and every walk $v_1\cdots v_{2t}$ the sequence $c(v_1) \cdots c(v_{2t})$ is not repetitive unless $v_i = v_{i+t}$ for every $i\in\{1,\ldots,t\}$ (resp. unless $v_i = v_{i+t}$ for some $i\in\{1,\ldots,t\}$). The walk (resp. stroll) chromatic number $\sigma(G)$ (resp. $\rho(G)$) of $G$ is the minimum $k$ for which $G$ has a walk-nonrepetitive (resp. stroll-nonrepetitive) $k$-coloring. Let $C_n$ and $P_n$ denote, respectively, the cycle and the path with $n$ vertices. In this paper we present three results that answer questions posed by Bar\'at and Wood in 2008: (i) $\sigma(C_n) = 4$ whenever $n\geq 4$ and $n \notin\{5,7\}$; (ii) $\rho(P_n) = 3$ if $3\leq n\leq 21$ and $\rho(P_n) = 4$ otherwise; and (iii) $\rho(C_n) = 4$, whenever $n \notin\{3,4,6,8\}$, and $\rho(C_n) = 3$ otherwise. In particular, (ii) improves bounds on $n$ obtained by Tao in 2023.

##### 7.Vector space Ramsey numbers and weakly Sidorenko affine configurations

**Authors:**Bryce Frederickson, Liana Yepremyan

**Abstract:** For $B \subseteq \mathbb F_q^m$, the $n$-th affine extremal number of $B$ is the maximum cardinality of a set $A \subseteq \mathbb F_q^n$ with no subset which is affinely isomorphic to $B$. Furstenberg and Katznelson proved that for any $B \subseteq \mathbb F_q^m$, the $n$-th affine extremal number of $B$ is $o(q^n)$ as $n \to \infty$. By counting affine homomorphisms between subsets of $\mathbb F_q^n$, we derive new bounds and give new proofs of some previously known bounds for certain affine extremal numbers. At the same time, we establish corresponding supersaturation results. We connect these bounds to certain Ramsey-type numbers in vector spaces over finite fields. For $s,t \geq 1$, let $R_q(s,t)$ denote the minimum $n$ such that in every red-blue coloring of the one-dimensional subspaces of $\mathbb F_q^n$, there is either a red $s$-dimensional subspace or a blue $t$-dimensional subspace of $\mathbb F_q^n$. The existence of these numbers is a special case of a well-known theorem of Graham, Leeb, Rothschild. We improve the best known upper bounds on $R_2(2,t)$, $R_3(2,t)$, $R_2(t,t)$, and $R_3(t,t)$.

##### 1.An interlacing property of the signless Laplacian of threshold graphs

**Authors:**Christoph Helmberg, Guilherme Porto, Guilherme Torres, Vilmar Trevisan

**Abstract:** We show that for threshold graphs, the eigenvalues of the signless Laplacian matrix interlace with the degrees of the vertices. As an application, we show that the signless Brouwer conjecture holds for threshold graphs, i.e., for threshold graphs the sum of the k largest eigenvalues is bounded by the number of edges plus k + 1 choose 2.

##### 2.Some orientation theorems for restricted DP-colorings of graphs

**Authors:**Ian Gossett

**Abstract:** We define signable, generalized signable, and $Z$-signable correspondence assignments on multigraphs, which generalize good correspondence assignments as introduced by Kaul and Mudrock. DP-colorings from these classes generalize signed colorings, signed $\mathbb{Z}_p$-colorings, and signed list colorings of signed graphs. We introduce an auxiliary digraph that allows us to prove an Alon-Tarsi style theorem for DP-colorings from $Z$-signable correspondence assignments on multigraphs, and obtain three DP-coloring analogs of the Alon-Tarsi theorem as corollaries.

##### 3.Sharp volume and multiplicity bounds for Fano simplices

**Authors:**Andreas Bäuerle

**Abstract:** We present sharp upper bounds on the volume, Mahler volume and multiplicity for Fano simplices depending on the dimension and Gorenstein index. These bounds rely on the interplay between lattice simplices and unit fraction partitions. Moreover, we present an efficient procedure for explicitly classifying Fano simplicies of any dimension and Gorenstein index and we carry out the classification up to dimension four for various Gorenstein indices.

##### 4.On a question of Matt Baker regarding the dollar game

**Authors:**Marine Cases-Thomas

**Abstract:** In an introductory paper on dollar game played on a graph, Matt Baker wrote the following: ``The total number of borrowing moves required to win the game when playing the 'borrowing binge strategy' is independent of which borrowing moves you do in which order! Note, however, that it is usually possible to win in fewer moves by employing lending moves in combination with borrowing moves. The optimal strategy when one uses both kinds of moves is not yet understood.'' In this article, we give a lower bound on the minimum number $ M_{\text{min}} $ of such moves of an optimal algorithm in terms of the number of moves $ M_0 $ of the borrowing binge strategy. Concretely, we have: $ M_{\text{min}} \geq \frac{M_0}{n-1} $ where $ n $ is the number of vertices of the graph. This bound is tight.

##### 5.Decreasing the mean subtree order by adding $k$ edges

**Authors:**Stijn Cambie, Guantao Chen, Yanli Hao, Nizamettin Tokar

**Abstract:** The mean subtree order of a given graph $G$, denoted $\mu(G)$, is the average number of vertices in a subtree of $G$. Let $G$ be a connected graph. Chin, Gordon, MacPhee, and Vincent [J. Graph Theory, 89(4): 413-438, 2018] conjectured that if $H$ is a proper spanning supergraph of $G$, then $\mu(H) > \mu(G)$. Cameron and Mol [J. Graph Theory, 96(3): 403-413, 2021] disproved this conjecture by showing that there are infinitely many pairs of graphs $H$ and $G$ with $H\supset G$, $V(H)=V(G)$ and $|E(H)|= |E(G)|+1$ such that $\mu(H) < \mu(G)$. They also conjectured that for every positive integer $k$, there exists a pair of graphs $G$ and $H$ with $H\supset G$, $V(H)=V(G)$ and $|E(H)| = |E(G)| +k$ such that $\mu(H) < \mu(G)$. Furthermore, they proposed that $\mu(K_m+nK_1) < \mu(K_{m, n})$ provided $n\gg m$. In this note, we confirm these two conjectures.

##### 6.On self-duality and unigraphicity for $3$-polytopes

**Authors:**Riccardo W. Maffucci

**Abstract:** Recent literature posed the problem of characterising the graph degree sequences with exactly one $3$-polytopal (i.e. planar, $3$-connected) realisation. This seems to be a difficult problem in full generality. In this paper, we characterise the sequences with exactly one self-dual $3$-polytopal realisation. An algorithm in the literature constructs a self-dual $3$-polytope for any admissible degree sequence. To do so, it performs operations on the radial graph, so that the corresponding $3$-polytope and its dual are modified in exactly the same way. To settle our question and construct the relevant graphs, we apply this algorithm, we introduce some modifications of it, and we also devise new ones. The speed of these algorithms is linear in the graph order.

##### 7.Approximate quadratic varieties

**Authors:**Luka Milićević

**Abstract:** A classical result in additive combinatorics, which is a combination of Balog-Szemer\'edi-Gowers theorem and a variant of Freiman's theorem due to Ruzsa, says that if a subset $A$ of $\mathbb{F}_p^n$ contains at least $c |A|^3$ additive quadruples, then there exists a subspace $V$, comparable in size to $A$, such that $|A \cap V| \geq \Omega_c(|A|)$. Motivated by the fact that higher order approximate algebraic structures play an important role in the theory of uniformity norms, it would be of interest to find higher order analogues of the mentioned result. In this paper, we study a quadratic version of the approximate property in question, namely what it means for a set to be an approximate quadratic variety. It turns out that information on the number of additive cubes, which are 8-tuples of the form $(x, x+ a,$ $x+ b, x+ c,$ $x+ a + b, x+ a + c,$ $x+ b + c, x+ a + b + c)$, in a set is insufficient on its own to guarantee quadratic structure, and it is necessary to restrict linear structure in a given set, which is a natural assumption in this context. With this in mind, we say that a subset $V$ of a finite vector space $G$ is a $(c_0, \delta, \varepsilon)$-approximate quadratic variety if $|V| = \delta |G|$, $\|1_V - \delta\|_{\mathsf{U}^2} \leq \varepsilon$ and $V$ contains at least $c_0\delta^7 |G|^4$ additive cubes. Our main result is the structure theorem for approximate quadratic varieties, stating that such a set has a large intersection with an exact quadratic variety of comparable size.

##### 1.Cliquewidth and dimension

**Authors:**Gwenaël Joret, Piotr Micek, Michał Pilipczuk, Bartosz Walczak

**Abstract:** We prove that every poset with bounded cliquewidth and with sufficiently large dimension contains the standard example of dimension $k$ as a subposet. This applies in particular to posets whose cover graphs have bounded treewidth, as the cliquewidth of a poset is bounded in terms of the treewidth of the cover graph. For the latter posets, we prove a stronger statement: every such poset with sufficiently large dimension contains the Kelly example of dimension $k$ as a subposet. Using this result, we obtain a full characterization of the minor-closed graph classes $\mathcal{C}$ such that posets with cover graphs in $\mathcal{C}$ have bounded dimension: they are exactly the classes excluding the cover graph of some Kelly example. Finally, we consider a variant of poset dimension called Boolean dimension, and we prove that posets with bounded cliquewidth have bounded Boolean dimension. The proofs rely on Colcombet's deterministic version of Simon's factorization theorem, which is a fundamental tool in formal language and automata theory, and which we believe deserves a wider recognition in structural and algorithmic graph theory.

##### 2.Combinatorial insight of Riemann Boundary value problem in lattice walk problems

**Authors:**Ruijie Xu

**Abstract:** Enumeration of quarter-plane lattice walks with small steps is a classical problem in combinatorics. An effective approach is the kernel method and the solution is derived by positive term extraction. Another approach is to reduce the lattice walk problem to a Carleman type Riemann boundary value problem (RBVP) and solve it by complex analysis. There are two parameter characterizing the solutions of a Carleman type Riemann Boundary value problem, the index $\chi$ and the conformal gluing function $w(x)$. In this paper, we propose a combinatorial insight to the RBVP approach. We show that the index can be treated as the canonical factorization in the kernel method and the conformal gluing function can be turned into a conformal mapping such that after this mapping, the positive degree terms and the negative degree terms can be naturally separated. The combinatorial insight of RBVP provide a connection between the kernel method, the RBVP approach and the Tutte's invariant method.

##### 3.Cellular diagonals of permutahedra

**Authors:**Bérénice Delcroix-Oger, Guillaume Laplante-Anfossi, Vincent Pilaud, Kurt Stoeckl

**Abstract:** We provide a systematic enumerative and combinatorial study of geometric cellular diagonals on the permutahedra. In the first part of the paper, we study the combinatorics of certain hyperplane arrangements obtained as the union of $\ell$ generically translated copies of the classical braid arrangement. Based on Zaslavsky's theory, we derive enumerative results on the faces of these arrangements involving combinatorial objects named partition forests and rainbow forests. This yields in particular nice formulas for the number of regions and bounded regions in terms of exponentials of generating functions of Fuss-Catalan numbers. By duality, the specialization of these results to the case $\ell = 2$ gives the enumeration of any geometric diagonal of the permutahedron. In the second part of the paper, we study diagonals which respect the operadic structure on the family of permutahedra. We show that there are exactly two such diagonals, which are moreover isomorphic. We describe their facets by a simple rule on paths in partition trees, and their vertices as pattern-avoiding pairs of permutations. We show that one of these diagonals is a topological enhancement of the Sanbeblidze-Umble diagonal, and unravel a natural lattice structure on their sets of facets. In the third part of the paper, we use the preceding results to show that there are precisely two isomorphic topological cellular operadic structures on the families of operahedra and multiplihedra, and exactly two infinity-isomorphic geometric universal tensor products of homotopy operads and A-infinity morphisms.

##### 4.On bicyclic graphs with maximal Graovac-Ghorbani index

**Authors:**Rui Song, Saihua Liu, Jianping Ou

**Abstract:** Graovac-Ghorbani index is a new version of the atom-bond connectivity index. D. Pacheco et al. [D. Pacheco, L. de Lima, C. S. Oliveira, On the Graovac-Ghorbani Index for Bicyclic Graph with No Pendent Vertices, MATCH Commun. Math. Comput. Chem. 86 (2021) 429-448] conjectured a sharp lower and upper bounds to the Graovac-Ghorbani index for all bicyclic graphs. Motivated by their nice work, in this paper we determine the maximal Graovac-Ghorbani index of bicyclic graphs and characterize the corresponding extremal graphs, which solves one of their Conjectures.

##### 5.Wreath Macdonald polynomials, a survey

**Authors:**Daniel Orr, Mark Shimozono

**Abstract:** Wreath Macdonald polynomials arise from the geometry of $\Gamma$-fixed loci of Hilbert schemes of points in the plane, where $\Gamma$ is a finite cyclic group of order $r\ge 1$. For $r=1$, they recover the classical (modified) Macdonald symmetric functions through Haiman's geometric realization of these functions. The existence, integrality, and positivity of wreath Macdonald polynomials for $r>1$ was conjectured by Haiman and first proved in work of Bezrukavnikov and Finkelberg by means of an equivalence of derived categories. Despite the power of this approach, a lack of explicit tools providing direct access to wreath Macdonald polynomials -- in the spirit of Macdonald's original works -- has limited progress in the subject. A recent result of Wen provides a remarkable set of such tools, packaged in the representation theory of quantum toroidal algebras. In this article, we survey Wen's result along with the basic theory of wreath Macdonald polynomials, including its geometric foundations and the role of bigraded reflection functors in the construction of wreath analogs of the $\nabla$ operator. We also formulate new conjectures on the values of important constants arising in the theory of wreath Macdonald $P$-polynomials. A variety of examples are used to illustrate these objects and constructions throughout the paper.

##### 6.Scaling limit of the sandpile identity element on the Sierpinski gasket

**Authors:**Robin Kaiser, Ecaterina Sava-Huss

**Abstract:** We investigate the identity element of the sandpile group on finite approximations of the Sierpinski gasket with normal boundary conditions and show that the sequence of piecewise constant continuations of the identity elements on SG_n converges in the weak* topology to the constant function with value 8/3 on the Sierpinski gasket SG. We then generalize the proof to a wider range of functions and obtain the scaling limit for the identity elements with different choices of sink vertices.

##### 7.The Hereditary Closure of the Unigraphs

**Authors:**Michael D. Barrus, Ann N. Trenk, Rebecca Whitman

**Abstract:** A graph with degree sequence $\pi$ is a \emph{unigraph} if it is isomorphic to every graph that has degree sequence $\pi$. The class of unigraphs is not hereditary and in this paper we study the related hereditary class HCU, the hereditary closure of unigraphs, consisting of all graphs induced in a unigraph. We characterize the class HCU in multiple ways making use of the tools of a decomposition due to Tyshkevich and a partial order on degree sequences due to Rao. We also provide a new characterization of the class that consists of unigraphs for which all induced subgraphs are also unigraphs.

##### 8.Cyclic Orderings of Paving Matroids

**Authors:**Sean McGuinness

**Abstract:** A matroid M is cyclically orderable if there is a cyclic permutation of the elements of M such that any r consecutive elements form a basis in M. An old conjecture of Kajitani, Miyano, and Ueno states that a matroid M is cyclically orderable if and only if for all nonempty subsets X in E(M), |X|/r(M) is less than or equal to |E(M)|/r(M). In this paper, we verify this conjecture for all paving matroids.

##### 9.Extremal, enumerative and probabilistic results on ordered hypergraph matchings

**Authors:**Michael Anastos, Zhihan Jin, Matthew Kwan, Benny Sudakov

**Abstract:** An ordered $r$-matching is an $r$-uniform hypergraph matching equipped with an ordering on its vertices. These objects can be viewed as natural generalisations of $r$-dimensional orders. The theory of ordered 2-matchings is well-developed and has connections and applications to extremal and enumerative combinatorics, probability, and geometry. On the other hand, in the case $r \ge 3$ much less is known, largely due to a lack of powerful bijective tools. Recently, Dudek, Grytczuk and Ruci\'nski made some first steps towards a general theory of ordered $r$-matchings, and in this paper we substantially improve several of their results and introduce some new directions of study. Many intriguing open questions remain.

##### 10.Assouad-Nagata dimension of minor-closed metrics

**Authors:**Chun-Hung Liu

**Abstract:** Assouad-Nagata dimension addresses both large and small scale behaviors of metric spaces and is a refinement of Gromov's asymptotic dimension. A metric space $M$ is a minor-closed metric if there exists an (edge)-weighted graph $G$ in a fixed minor-closed family such that the underlying space of $M$ is the vertex-set of $G$, and the metric of $M$ is the distance function in $G$. Minor-closed metrics naturally arise when removing redundant edges of the underlying graphs by using edge-deletion and edge-contraction. In this paper, we determine the Assouad-Nagata dimension of every minor-closed metric. It is a common generalization of known results for the asymptotic dimension of $H$-minor free unweighted graphs and the Assouad-Nagata dimension of some 2-dimensional continuous spaces (e.g.\ complete Rienmannian surfaces with finite Euler genus) and their corollaries.

##### 11.Tiling Dense Hypergraphs

**Authors:**Richard Lang

**Abstract:** In the perfect tiling problem, we aim to cover the vertices of a hypergraph $G$ with pairwise vertex-disjoint copies of a hypergraph $F$. There are three essentially necessary conditions for such a perfect tiling, which correspond to barriers in space, divisibility and covering. It is natural to ask in which situations these conditions are also asymptotically sufficient. Our main result confirms this for all hypergraph families that are approximately closed under taking typical induced subgraphs of constant order. Among others, this includes families parametrised by minimum degrees and quasirandomness, which have been studied extensively in this area. As an application, we recover and extend a series of well-known results for perfect tilings in hypergraphs and related settings involving vertex orderings and rainbow structures.

##### 1.Approximate Core Allocations for Edge Cover Games

**Authors:**Tianhang Lu, Han Xian, Qizhi Fang

**Abstract:** We study the approximate core for edge cover games, which are cooperative games stemming from edge cover problems. In these games, each player controls a vertex on a network $G = (V, E; w)$, and the cost of a coalition $S\subseteq V$ is equivalent to the minimum weight of edge covers in the subgraph induced by $S$. We prove that the 3/4-core of edge cover games is always non-empty and can be computed in polynomial time by using linear program duality approach. This ratio is the best possible, as it represents the integrality gap of the natural LP for edge cover problems. Moreover, our analysis reveals that the ratio of approximate core corresponds with the length of the shortest odd cycle of underlying graphs.

##### 2.Intersection subgroup graph with forbidden subgraphs

**Authors:**Santanu Mandal, Pallabi Manna

**Abstract:** Let $G$ be a group. The intersection subgroup graph of $G$ (introduced by Anderson et al. \cite{anderson}) is the simple graph $\Gamma_{S}(G)$ whose vertices are those non-trivial subgroups say $H$ of $G$ with $H\cap K=\{e\}$ for some non-trivial subgroup $K$ of $G$; two distinct vertices $H$ and $K$ are adjacent if and only if $H\cap K=\{e\}$, where $e$ is the identity element of $G$. In this communication, we explore the groups whose intersection subgroup graph belongs to several significant graph classes including cluster graphs, perfect graphs, cographs, chordal graphs, bipartite graphs, triangle-free and claw-fee graphs. We categorize each nilpotent group $G$ so that $\Gamma_S(G)$ belongs to the above classes. We entirely classify the simple group of Lie type whose intersection subgroup graph is a cograph. Moreover, we deduce that $\Gamma_{S}(G)$ is neither a cograph nor a chordal graph if $G$ is a torsion-free nilpotent group.

##### 3.A class of graphs of zero Turán density in a hypercube

**Authors:**Maria Axenovich

**Abstract:** A graph is cubical if it is a subgraph of a hypercube. For a cubical graph $H$ and a hypercube $Q_n$, $ex(Q_n, H)$ is the largest number of edges in an $H$-free subgraph of $Q_n$. If $ex(Q_n, H)$ is at least a positive proportion of the number of edges in $Q_n$, $H$ is said to have a positive Tur\'an density in a hypercube or simply a positive Tur\'an density; otherwise it has a zero Tur\'an density. Determining $ex(Q_n, H)$ and even identifying whether $H$ has a positive or a zero Tur\'an density remains a widely open question for general $H$. By relating extremal numbers in a hypercube and certain corresponding hypergraphs, Conlon found a large class of cubical graphs, ones having so-called partite representation, that have a zero Tur\'an density. He raised a question whether this gives a characterisation, i.e., whether a cubical graph has zero Tur\'an density if and only if it has partite representation. Here, we show that, as suspected by Conlon, this is not the case. We give an example of a class of cubical graphs which have no partite representation, but on the other hand, have a zero Tur\'an density. In addition, we show that any graph whose every block has partite representation has a zero Tur\'an density in a hypercube.

##### 4.On the subgroup regular set in Cayley graphs

**Authors:**Asamin Khaefi, Zeinab Akhlaghi, Behrooz Khosravi

**Abstract:** A subset $C$ of the vertex set of a graph $\Gamma$ is said to be $(a,b)$-regular if $C$ induces an $a$-regular subgraph and every vertex outside $C$ is adjacent to exactly $b$ vertices in $C$. In particular, if $C$ is an $(a,b)$-regular set of some Cayley graph on a finite group $G$, then $C$ is called an $(a,b)$-regular set of $G$ and a $(0,1)$-regular set is called a perfect code of $G$. In [Wang, Xia and Zhou, Regular sets in Cayley graphs, J. Algebr. Comb., 2022] it is proved that if $H$ is a normal subgroup of $G$, then $H$ is a perfect code of $G$ if and only if it is an $(a,b)$-regular set of $G$, for each $0\leq a\leq|H|-1$ and $0\leq b\leq|H|$ with $\gcd(2,|H|-1)\mid a$. In this paper, we generalize this result and show that a subgroup $H$ of $G$ is a perfect code of $G$ if and only if it is an $(a,b)$-regular set of $G$, for each $0\leq a\leq|H|-1$ and $0\leq b\leq|H|$ such that $\gcd(2,|H|-1)$ divides $a$.

##### 5.Zero Forcing on 2-connected Outerplanar Graphs

**Authors:**Nolan Ison, Mark Kempton, Franklin Kenter

**Abstract:** We determine upper and lower bounds on the zero forcing number of 2-connected outerplanar graphs in terms of the structure of the weak dual. We show that the upper bound is always at most half the number of vertices of the graph. This work generalizes work of Hern\'andez, Ranilla and Ranilla-Cortina who proved a similar result for maximal outerplanar graphs.

##### 6.Neumaier Cayley graphs

**Authors:**Mojtaba Jazaeri

**Abstract:** A Neumaier graph is a non-complete edge-regular graph with the property that it has a regular clique. In this paper, we study Neumaier Cayley graphs. We give a necessary and sufficient condition under which a Neumaier Cayley graph is a strongly regular Neumaier Cayley graph. We also characterize Neumaier Cayley graphs with small valency at most $10$.

##### 7.Graph-like Scheduling Problems and Property B

**Authors:**John Machacek

**Abstract:** Breuer and Klivans defined a diverse class of scheduling problems in terms of Boolean formulas with atomic clauses that are inequalities. We consider what we call graph-like scheduling problems. These are Boolean formulas that are conjunctions of disjunctions of atomic clauses $(x_i \neq x_j)$. These problems generalize proper coloring in graphs and hypergraphs. We focus on the existence of a solution with all $x_i$ taking the value of $0$ or $1$ (i.e. problems analogous to the bipartite case). When a graph-like scheduling problem has such a solution, we say it has property B just as is done for $2$-colorable hypergraphs. We define the notion of a $\lambda$-uniform graph-like scheduling problem for any integer partition $\lambda$. Some bounds are attained for the size of the smallest $\lambda$-uniform graph-like scheduling problems without property B. We make use of both random and constructive methods to obtain bounds. Just as in the case of hypergraphs finding tight bounds remains an open problem.

##### 8.Ascending Subgraph Decomposition

**Authors:**Kyriakos Katsamaktsis, Shoham Letzter, Alexey Pokrovskiy, Benny Sudakov

**Abstract:** A typical theme for many well-known decomposition problems is to show that some obvious necessary conditions for decomposing a graph $G$ into copies $H_1, \ldots, H_m$ are also sufficient. One such problem was posed in 1987, by Alavi, Boals, Chartrand, Erd\H{o}s, and Oellerman. They conjectured that the edges of every graph with $\binom{m+1}2$ edges can be decomposed into subgraphs $H_1, \dots, H_m$ such that each $H_i$ has $i$ edges and is isomorphic to a subgraph of $H_{i+1}$. In this paper we prove this conjecture for sufficiently large $m$.

##### 1.Upper bounds of dual flagged Weyl characters

**Authors:**Zhuowei Lin, Simon C. Y. Peng, Sophie C. C. Sun

**Abstract:** For a subset $D$ of boxes in an $n\times n$ square grid, let $\chi_{D}(x)$ denote the dual character of the flagged Weyl module associated to $D$. It is known that $\chi_{D}(x)$ specifies to a Schubert polynomial (resp., a key polynomial) in the case when $D$ is the Rothe diagram of a permutation (resp., the skyline diagram of a composition). One can naturally define a lower and an upper bound of $\chi_{D}(x)$. M{\'e}sz{\'a}ros, St. Dizier and Tanjaya conjectured that $\chi_{D}(x)$ attains the upper bound if and only if $D$ avoids a certain subdiagram. We provide a proof of this conjecture.

##### 2.Stabbing boxes with finitely many axis-parallel lines and flats

**Authors:**Sutanoya Chakraborty, Arijit Ghosh, Soumi Nandi

**Abstract:** We give necessary and sufficient condition for an infinite collection of axis-parallel boxes in $\mathbb{R}^{d}$ to be pierceable by finitely many axis-parallel $k$-flats, where $0 \leq k < d$. We also consider colorful generalizations of the above result and establish their feasibility. The problem considered in this paper is an infinite variant of the Hadwiger-Debrunner $(p,q)$-problem.

##### 3.Dimension Independent Helly Theorem for Lines and Flats

**Authors:**Sutanoya Chakraborty, Arijit Ghosh, Soumi Nandi

**Abstract:** We give a generalization of dimension independent Helly Theorem of Adiprasito, B\'{a}r\'{a}ny, Mustafa, and Terpai (Discrete & Computational Geometry 2022) to higher dimensional transversal. We also prove some impossibility results that establish the tightness of our extension.

##### 4.Ramsey numbers of color critical graphs versus large generalized fans

**Authors:**Taiping Jiang, Xinmin Hou

**Abstract:** Given two graphs $G$ and $H$, the {Ramsey number} $R(G,H)$ is the smallest positive integer $N$ such that every 2-coloring of the edges of $K_{N}$ contains either a red $G$ or a blue $H$. Let $K_{N-1}\sqcup K_{1,k}$ be the graph obtained from $K_{N-1}$ by adding a new vertex $v$ connecting $k$ vertices of $K_{N-1}$. Hook and Isaak (2011) defined the {\em star-critical Ramsey number} $r_{*}(G,H)$ as the smallest integer $k$ such that every 2-coloring of the edges of $K_{N-1}\sqcup K_{1,k}$ contains either a red $G$ or a blue $H$, where $N=R(G, H)$. For sufficiently large $n$, Li and Rousseau~(1996) proved that $R(K_{k+1},K_{1}+nK_{t})=knt +1$, Hao, Lin~(2018) showed that $r_{*}(K_{k+1},K_{1}+nK_{t})=(k-1)tn+t$; Li and Liu~(2016) proved that $R(C_{2k+1}, K_{1}+nK_{t})=2nt+1$, and Li, Li, and Wang~(2020) showed that $r_{*}(C_{2m+1},K_{1}+nK_{t})=nt+t$. A graph $G$ with $\chi(G)=k+1$ is called edge-critical if $G$ contains an edge $e$ such that $\chi(G-e)=k$. In this paper, we extend the above results by showing that for an edge-critical graph $G$ with $\chi(G)=k+1$, when $k\geq 2$, $t\geq 2$ and $n$ is sufficiently large, $R(G, K_{1}+nK_{t})=knt+1$ and $r_{*}(G,K_{1}+nK_{t})=(k-1)nt+t$.

##### 5.On the super edge-magicness of graphs with a specific degree sequence

**Authors:**Rikio Ichishima, Francesc A. Muntaner-Batle

**Abstract:** A graph $G$ is said to be super edge-magic if there exists a bijective function $f:V\left(G\right) \cup E\left(G\right)\rightarrow \left\{1, 2, \ldots , \left\vert V\left( G\right) \right\vert +\left\vert E\left( G\right) \right\vert \right\}$ such that $f\left(V \left(G\right)\right) =\left\{1, 2, \ldots , \left\vert V\left( G\right) \right\vert \right\}$ and $f\left(u\right) + f\left(v\right) + f\left(uv\right)$ is a constant for each $uv\in E\left( G\right) $. In this paper, we study the super edge-magicness of graphs of order $n$ with degree sequence $s:4, 2, 2, \ldots, 2$. We also investigate the super edge-magic properties of certain families of graphs. This leads us to propose some open problems.

##### 6.Some applications of linear algebraic methods in combinatorics

**Authors:**Maryam Khosravi, Ebadollah S. Mahmoodian

**Abstract:** In this note, we intend to produce all latin squares from one of them using suitable move which is defined by small trades and do the similar work on 4-cycle systems. These problems, reformulate as finding basis for the kernel of special matrices, representef to some graphs.

##### 7.On a method of Alweiss

**Authors:**Zach Hunter

**Abstract:** Recently, Alweiss settled Hindman's conjecture over the rationals. In this paper, we provide our own exposition of Alweiss' result, and show how to modify his method to also show that sums of distinct products are partition regular over the rationals.

##### 8.Computing Optimal Leaf Roots of Chordal Cographs in Linear Time

**Authors:**Van Bang Le, Christian Rosenke

**Abstract:** A graph G is a k-leaf power, for an integer k >= 2, if there is a tree T with leaf set V(G) such that, for all vertices x, y in V(G), the edge xy exists in G if and only if the distance between x and y in T is at most k. Such a tree T is called a k-leaf root of G. The computational problem of constructing a k-leaf root for a given graph G and an integer k, if any, is motivated by the challenge from computational biology to reconstruct phylogenetic trees. For fixed k, Lafond [SODA 2022] recently solved this problem in polynomial time. In this paper, we propose to study optimal leaf roots of graphs G, that is, the k-leaf roots of G with minimum k value. Thus, all k'-leaf roots of G satisfy k <= k'. In terms of computational biology, seeking optimal leaf roots is more justified as they yield more probable phylogenetic trees. Lafond's result does not imply polynomial-time computability of optimal leaf roots, because, even for optimal k-leaf roots, k may (exponentially) depend on the size of G. This paper presents a linear-time construction of optimal leaf roots for chordal cographs (also known as trivially perfect graphs). Additionally, it highlights the importance of the parity of the parameter k and provides a deeper insight into the differences between optimal k-leaf roots of even versus odd k. Keywords: k-leaf power, k-leaf root, optimal k-leaf root, trivially perfect leaf power, chordal cograph

##### 9.Ramsey numbers of hypergraphs of a given size

**Authors:**Domagoj Bradač, Jacob Fox, Benny Sudakov

**Abstract:** The $q$-color Ramsey number of a $k$-uniform hypergraph $H$ is the minimum integer $N$ such that any $q$-coloring of the complete $k$-uniform hypergraph on $N$ vertices contains a monochromatic copy of $H$. The study of these numbers is one of the central topics in Combinatorics. In 1973, Erd\H{o}s and Graham asked to maximize the Ramsey number of a graph as a function of the number of its edges. Motivated by this problem, we study the analogous question for hypergaphs. For fixed $k \ge 3$ and $q \ge 2$ we prove that the largest possible $q$-color Ramsey number of a $k$-uniform hypergraph with $m$ edges is at most $\mathrm{tw}_k(O(\sqrt{m})),$ where $\mathrm{tw}$ denotes the tower function. We also present a construction showing that this bound is tight for $q \ge 4$. This resolves a problem by Conlon, Fox and Sudakov. They previously proved the upper bound for $k \geq 4$ and the lower bound for $k=3$. Although in the graph case the tightness follows simply by considering a clique of appropriate size, for higher uniformities the construction is rather involved and is obtained by using paths in expander graphs.

##### 10.Generalized point configurations in ${\mathbb F}_q^d$

**Authors:**Paige Bright, Xinyu Fang, Barrett Heritage, Alex Iosevich, Tingsong Jiang, Hans Parshall, Maxwell Sun

**Abstract:** In this paper, we generalize \cite{IosevichParshall}, \cite{LongPaths} and \cite{cycles} by allowing the \emph{distance} between two points in a finite field vector space to be defined by a general non-degenerate bilinear form or quadratic form. We prove the same bounds on the sizes of large subsets of $\F_q^d$ for them to contain distance graphs with a given maximal vertex degree, under the more general notion of distance. We also prove the same results for embedding paths, trees and cycles in the general setting.

##### 11.A study of $4-$cycle systems

**Authors:**B. Bagheri Gh., M. Khosravi, E. S. Mahmoodian, S. Rashidi

**Abstract:** A $4-$cycle system is a partition of the edges of the complete graph $K_n$ into $4-$cycles. Let ${ C}$ be a collection of cycles of length 4 whose edges partition the edges of $K_n$. A set of 4-cycles $T_1 \subset C$ is called a 4-cycle trade if there exists a set $T_2$ of edge-disjoint 4-cycles on the same vertices, such that $({C} \setminus T_1)\cup T_2$ also is a collection of cycles of length 4 whose edges partition the edges of $K_n$. We study $4-$cycle trades of volume two (double-diamonds) and three and show that the set of all 4-CS(9) is connected with respect of trading with trades of volume 2 (double-diamond) and 3. In addition, we present a full rank matrix whose null-space is containing trade-vectors.

##### 1.Improved bounds for embedding certain configurations in subsets of vector spaces over finite fields

**Authors:**Paige Bright, Xinyu Fang, Barrett Heritage, Alex Iosevich, Maxwell Sun

**Abstract:** The fourth listed author and Hans Parshall (\cite{IosevichParshall}) proved that if $E \subset {\mathbb F}_q^d$, $d \ge 2$, and $G$ is a connected graph on $k+1$ vertices such that the largest degree of any vertex is $m$, then if $|E| \ge C q^{m+\frac{d-1}{2}}$, for any $t>0$, there exist $k+1$ points $x^1, \dots, x^{k+1}$ in $E$ such that $||x^i-x^j||=t$ if the $i$'th vertex is connected to the $j$'th vertex by an edge in $G$. In this paper, we give several indications that the maximum degree is not always the right notion of complexity and prove several concrete results to obtain better exponents than the Iosevich-Parshall result affords. This can be viewed as a step towards understanding the right notion of complexity for graph embeddings in subsets of vector spaces over finite fields.

##### 1.Slitherlink Signatures

**Authors:**Nikolai Beluhov

**Abstract:** Let $G$ be a planar graph and let $C$ be a cycle in $G$. Inside of each finite face of $G$, we write down the number of edges of that face which belong to $C$. This is the signature of $C$ in $G$. The notion of a signature arises naturally in the context of Slitherlink puzzles. The signature of a cycle does not always determine it uniquely. We focus on the ambiguity of signatures in the case when $G$ is a rectangular grid of unit square cells. We describe all grids which admit an ambiguous signature. For each such grid, we then determine the greatest possible difference between two cycles with the same signature on it. We also study the possible values of the total number of cycles which fit a given signature. We discuss various related questions as well.

##### 2.Monochromatic infinite sets in Minkowski planes

**Authors:**Nóra Frankl, Panna Gehér, Arsenii Sagdeev, Géza Tóth

**Abstract:** We prove that for any $\ell_p$-norm in the plane with $1<p<\infty$ and for every infinite $\mathcal{M} \subset \mathbb{R}^2$, there exists a two-colouring of the plane such that no isometric copy of $\mathcal{M}$ is monochromatic. On the contrary, we show that for every polygonal norm (that is, the unit ball is a polygon) in the plane, there exists an infinite $\mathcal{M} \subset \mathbb{R}^2$ such that for every two-colouring of the plane there exists a monochromatic isometric copy of $\mathcal{M}$.

##### 3.Maximum chordal subgraphs of random graphs

**Authors:**Michael Krivelevich, Maksim Zhukovskii

**Abstract:** We find asymptotics of the maximum size of a chordal subgraph in a binomial random graph $G(n,p)$, for $p=\mathrm{const}$ and $p=n^{-\alpha+o(1)}$.

##### 4.Weakly and Strongly Fan-Planar Graphs

**Authors:**Otfried Cheong, Henry Förster, Julia Katheder, Maximilian Pfister, Lena Schlipf

**Abstract:** We study two notions of fan-planarity introduced by (Cheong et al., GD22), called weak and strong fan-planarity that separate two non-equivalent definitions of fan-planarity in the literature. We prove that not every weakly fan-planar graph is strongly fan-planar, while the density upper bound for both families is the same.

##### 5.Geodetic Graphs: Experiments and New Constructions

**Authors:**Florian Stober, Armin Weiß

**Abstract:** In 1962 Ore initiated the study of geodetic graphs. A graph is called geodetic if the shortest path between every pair of vertices is unique. In the subsequent years a wide range of papers appeared investigating their peculiar properties. Yet, a complete classification of geodetic graphs is out of reach. In this work we present a program enumerating all geodetic graphs of a given size. Using our program, we succeed to find all geodetic graphs with up to 25 vertices and all regular geodetic graphs with up to 32 vertices. This leads to the discovery of two new infinite families of geodetic graphs.

##### 6.Characterizing bipartite distance-regularized graphs with vertices of eccentricity 4

**Authors:**Blas Fernández, Marija Maksimović, Sanja Rukavina

**Abstract:** The characterization of bipartite distance-regularized graphs, where some vertices have eccentricity less than four, in terms of the incidence structures of which they are incidence graphs, is known. In this paper we prove that there is a one-to-one correspondence between the incidence graphs of quasi-symmetric SPBIBDs with parameters $(v,b,r,k, \lambda_1,0)$ of type $(k-1,t)$ with intersection numbers $x=0$ and $y>0$, where $0< y\leq t<k$ , and bipartite distance-regularized graphs with $D=D'=4$.

##### 7.Sparse groups need not be semisparse

**Authors:**Isabel Hubard, Micael Toledo

**Abstract:** In 1999 Michael Hartley showed that any abstract polytope can be constructed as a double coset poset, by means of a C-group $\C$ and a subgroup $N \leq \C$. Subgroups $N \leq \C$ that give rise to abstract polytopes through such construction are called {\em sparse}. If, further, the stabilizer of a base flag of the poset is precisely $N$, then $N$ is said to be {\em semisparse}. In \cite[Conjecture 5.2]{hartley1999more} Hartley conjectures that sparse groups are always semisparse. In this paper, we show that this conjecture is in fact false: there exist sparse groups that are not semisparse. In particular, we show that such groups are always obtained from non-faithful maniplexes that give rise to polytopes. Using this, we show that Hartely's conjecture holds for rank 3, but we construct examples to disprove the conjecture for all ranks $n\geq 4$.

##### 8.The Cyclic Cutwidth of $Q_n$

**Authors:**Jason Erbele, Joseph D. Chavez, Rolland Trapp

**Abstract:** In this article the cyclic cutwidth of the $n$-dimensional cube is explored. It has been conjectured by Dr. Chavez and Dr. Trapp that the cyclic cutwidth of $Q_n$ is minimized with the Graycode numbering. Several results have been found toward the proof of this conjecture.

##### 9.The planar Turán number of $\{K_4,C_5\}$ and $\{K_4,C_6\}$

**Authors:**Ervin Győri, Alan Li, Runtian Zhou

**Abstract:** Let $\mathcal{H}$ be a set of graphs. The planar Tur\'an number, $ex_\mathcal{P}(n,\mathcal{H})$, is the maximum number of edges in an $n$-vertex planar graph which does not contain any member of $\mathcal{H}$ as a subgraph. When $\mathcal{H}=\{H\}$ has only one element, we usually write $ex_\mathcal{P}(n,H)$ instead. The topic of extremal planar graphs was initiated by Dowden (2016). He obtained sharp upper bound for both $ex_\mathcal{P}(n,C_5)$ and $ex_\mathcal{P}(n,K_4)$. Later on, we obtained sharper bound for $ex_\mathcal{P}(n,\{K_4,C_7\})$. In this paper, we give upper bounds of $ex_\mathcal{P}(n,\{K_4,C_5\})\leq {15\over 7}(n-2)$ and $ex_\mathcal{P}(n,\{K_4,C_6\})\leq {7\over 3}(n-2)$. We also give constructions which show the bounds are tight for infinitely many graphs.

##### 1.On graphs with no induced $P_5$ or $K_5-e$

**Authors:**Arnab Char, T. Karthick

**Abstract:** In this paper, we are interested in some problems related to chromatic number and clique number for the class of $(P_5,K_5-e)$-free graphs, and prove the following. $(a)$ If $G$ is a connected ($P_5,K_5-e$)-free graph with $\omega(G)\geq 7$, then either $G$ is the complement of a bipartite graph or $G$ has a clique cut-set. Moreover, there is a connected ($P_5,K_5-e$)-free imperfect graph $H$ with $\omega(H)=6$ and has no clique cut-set. This strengthens a result of Malyshev and Lobanova [Disc. Appl. Math. 219 (2017) 158--166]. $(b)$ If $G$ is a ($P_5,K_5-e$)-free graph with $\omega(G)\geq 4$, then $\chi(G)\leq \max\{7, \omega(G)\}$. Moreover, the bound is tight when $\omega(G)\notin \{4,5,6\}$. This result together with known results partially answers a question of Ju and Huang [arXiv:2303.18003 [math.CO] 2023], and also improves a result of Xu [Manuscript 2022]. While the "Chromatic Number Problem" is known to be $NP$-hard for the class of $P_5$-free graphs, our results together with some known results imply that the "Chromatic Number Problem" can be solved in polynomial time for the class of ($P_5,K_5-e$)-free graphs which may be independent interest.

##### 2.The maximum four point condition matrix of a tree

**Authors:**Ali Azimi, Rakesh Jana, Mukesh Kumar Nagar, Sivaramakrishnan Sivasubramanian

**Abstract:** $\newcommand{\Max}{\mathrm{Max4PC}}$ The Four point condition (4PC henceforth) is a well known condition characterising distances in trees $T$. Let $w,x,y,z$ be four vertices in $T$ and let $d_{x,y}$ denote the distance between vertices $x,y$ in $T$. The 4PC condition says that among the three terms $d_{w,x} + d_{y,z}$, $d_{w,y} + d_{x,z}$ and $d_{w,z} + d_{x,y}$ the maximum value equals the second maximum value. We define an $\binom{n}{2} \times \binom{n}{2}$ sized matrix $\Max_T$ from a tree $T$ where the rows and columns are indexed by size-2 subsets. The entry of $\Max_T$ corresponding to the row indexed by $\{w,x\}$ and column $\{y,z\}$ is the maximum value among the three terms $d_{w,x} + d_{y,z}$, $d_{w,y} + d_{x,z}$ and $d_{w,z} + d_{x,y}$. In this work, we determine basic properties of this matrix like rank, give an algorithm that outputs a family of bases, and find the determinant of $\Max_T$ when restricted to our basis. We further determine the inertia and the Smith Normal Form (SNF) of $\Max_T$.

##### 3.Boundes for Boxicity of some classes of graphs

**Authors:**T. Kavaskar

**Abstract:** Let $box(G)$ be the boxicity of a graph $G$, $G[H_1,H_2,\ldots, H_n]$ be the $G$-generalized join graph of $n$-pairwise disjoint graphs $H_1,H_2,\ldots, H_n$, $G^d_k$ be a circular clique graph (where $k\geq 2d$) and $\Gamma(R)$ be the zero-divisor graph of a commutative ring $R$. In this paper, we prove that $\chi(G^d_k)\geq box(G^d_k)$, for all $k$ and $d$ with $k\geq 2d$. This generalizes the results proved in \cite{Aki}. Also we obtain that $box(G[H_1,H_2,\ldots,H_n])\leq \mathop\sum\limits_{i=1}^nbox(H_i)$. As a consequence of this result, we obtain a bound for boxicity of zero-divisor graph of a finite commutative ring with unity. In particular, if $R$ is a finite commutative non-zero reduced ring with unity, then $\chi(\Gamma(R))\leq box(\Gamma(R))\leq 2^{\chi(\Gamma(R))}-2$. where $\chi(\Gamma(R))$ is the chromatic number of $\Gamma(R)$. Moreover, we show that if $N= \prod\limits_{i=1}^{a}p_i^{2n_i} \prod\limits_{j=1}^{b}q_j^{2m_j+1}$ is a composite number, where $p_i$'s and $q_j$'s are distinct prime numbers, then $box(\Gamma(\mathbb{Z}_N))\leq \big(\mathop\prod\limits_{i=1}^{a}(2n_i+1)\mathop\prod\limits_{j=1}^{b}(2m_j+2)\big)-\big(\mathop\prod\limits_{i=1}^{a}(n_i+1)\mathop\prod\limits_{j=1}^{b}(m_j+1)\big)-1$, where $\mathbb{Z}_N$ is the ring of integers modulo $N$. Further, we prove that, $box(\Gamma(\mathbb{Z}_N))=1$ if and only if either $N=p^n$ for some prime number $p$ and some positive integer $n\geq 2$ or $N=2p$ for some odd prime number $p$.

##### 4.Cameron-Liebler sets in permutation groups

**Authors:**Jozefien D'haeseleer, Karen Meagher, Venkata Raghu Tej Pantangi

**Abstract:** Consider a group $G$ acting on a set $\Omega$, the vector $v_{a,b}$ is a vector with the entries indexed by the elements of $G$, and the $g$-entry is 1 if $g$ maps $a$ to $b$, and zero otherwise. A $(G,\Omega)$-Cameron-Liebler set is a subset of $G$, whose indicator function is a linear combination of elements in $\{v_{a, b}\ :\ a, b \in \Omega\}$. We investigate Cameron-Liebler sets in permutation groups, with a focus on constructions of Cameron-Liebler sets for 2-transitive groups.

##### 5.Asymptotic behavior of Laplacian eigenvalues of subspace inclusion graphs

**Authors:**Alan Lew

**Abstract:** Let $\text{Fl}_{n,q}$ be the simplicial complex whose vertices are the non-trivial subspaces of $\mathbb{F}_q^n$ and whose simplices correspond to families of subspaces forming a flag. Let $\Delta^{+}_k(\text{Fl}_{n,q})$ be the $k$-dimensional weighted upper Laplacian on $ \text{Fl}_{n,q}$. The spectrum of $\Delta^{+}_k(\text{Fl}_{n,q})$ was first studied by Garland, who obtained a lower bound on its non-zero eigenvalues. Here, we focus on the $k=0$ case. We determine the asymptotic behavior of the eigenvalues of $\Delta_{0}^{+}(\text{Fl}_{n,q})$ as $q$ tends to infinity. In particular, we show that for large enough $q$, $\Delta_{0}^{+}(\text{Fl}_{n,q})$ has exactly $\left\lfloor n^2/4\right\rfloor+2$ distinct eigenvalues, and that every eigenvalue $\lambda\neq 0,n-1$ of $\Delta_{0}^{+}(\text{Fl}_{n,q})$ tends to $n-2$ as $q$ goes to infinity. This solves the $0$-dimensional case of a conjecture of Papikian.

##### 6.Optimal spread for spanning subgraphs of Dirac hypergraphs

**Authors:**Tom Kelly, Alp Müyesser, Alexey Pokrovskiy

**Abstract:** Let $G$ and $H$ be hypergraphs on $n$ vertices, and suppose $H$ has large enough minimum degree to necessarily contain a copy of $G$ as a subgraph. We give a general method to randomly embed $G$ into $H$ with good "spread". More precisely, for a wide class of $G$, we find a randomised embedding $f\colon G\hookrightarrow H$ with the following property: for every $s$, for any partial embedding $f'$ of $s$ vertices of $G$ into $H$, the probability that $f$ extends $f'$ is at most $O(1/n)^s$. This is a common generalisation of several streams of research surrounding the classical Dirac-type problem. For example, setting $s=n$, we obtain an asymptotically tight lower bound on the number of embeddings of $G$ into $H$. This recovers and extends recent results of Glock, Gould, Joos, K\"uhn, and Osthus and of Montgomery and Pavez-Sign\'e regarding enumerating Hamilton cycles in Dirac hypergraphs. Moreover, using the recent developments surrounding the Kahn--Kalai conjecture, this result implies that many Dirac-type results hold robustly, meaning $G$ still embeds into $H$ after a random sparsification of its edge set. This allows us to recover a recent result of Kang, Kelly, K\"uhn, Osthus, and Pfenninger and of Pham, Sah, Sawhney, and Simkin for perfect matchings, and obtain novel results for Hamilton cycles and factors in Dirac hypergraphs. Notably, our randomised embedding algorithm is self-contained and does not require Szemer\'edi's regularity lemma or iterative absorption.

##### 1.Extremal problems for disjoint graphs

**Authors:**Zhenyu Ni, Jing Wang, Liying Kang

**Abstract:** For a simple graph $F$, let $\mathrm{EX}(n, F)$ and $\mathrm{EX_{sp}}(n,F)$ be the set of graphs with the maximum number of edges and the set of graphs with the maximum spectral radius in an $n$-vertex graph without any copy of the graph $F$, respectively. Let $F$ be a graph with $\mathrm{ex}(n,F)=e(T_{n,r})+O(1)$. In this paper, we show that $\mathrm{EX_{sp}}(n,kF)\subseteq \mathrm{EX}(n,kF)$ for sufficiently large $n$. This generalizes a result of Wang, Kang and Xue [J. Comb. Theory, Ser. B, 159(2023) 20-41]. We also determine the extremal graphs of $kF$ in term of the extremal graphs of $F$.

##### 2.Some experimental observations about Hankel determinants of convolution powers of Catalan numbers

**Authors:**Johann Cigler

**Abstract:** Computer experiments suggest some conjectures about Hankel determinants of convolution powers of Catalan numbers. Unfortunately, for most of them I have no proofs. I would like to present them anyway hoping that someone finds them interesting and can prove them.

##### 3.Connectivity Graph-Codes

**Authors:**Noga Alon

**Abstract:** The symmetric difference of two graphs $G_1,G_2$ on the same set of vertices $V$ is the graph on $V$ whose set of edges are all edges that belong to exactly one of the two graphs $G_1,G_2$. For a fixed graph $H$ call a collection ${\cal G}$ of spanning subgraphs of $H$ a connectivity code for $H$ if the symmetric difference of any two distinct subgraphs in ${\cal G}$ is a connected spanning subgraph of $H$. It is easy to see that the maximum possible cardinality of such a collection is at most $2^{k'(H)} \leq 2^{\delta(H)}$, where $k'(H)$ is the edge-connectivity of $H$ and $\delta(H)$ is its minimum degree. We show that equality holds for any $d$-regular (mild) expander, and observe that equality does not hold in several natural examples including powers of long cycles and products of a small clique with a long cycle.

##### 4.Ramsey-type results on parameters related to domination

**Authors:**Jin Sun

**Abstract:** There is a philosophy to discover Ramsey-type theorem: given a graph parameter $\mu$, characterize the family $\HH$ of graphs which satisfies that every $\HH$-free graph $G$ has bounded parameter $\mu$. The classical Ramsey's theorem deals the parameter $\mu$ as the number of vertices. It also has a corresponding connected version. This Ramsey-type problem on domination number has been solved by Furuya. We will use this result to handle more parameters related to domination.

##### 5.Uniquely Distinguishing Colorable Graphs

**Authors:**M. Korivand, N. Soltankhah, K. Khashyarmanesh

**Abstract:** A graph is called uniquely distinguishing colorable if there is only one partition of vertices of the graph that forms distinguishing coloring with the smallest possible colors. In this paper, we study the unique colorability of the distinguishing coloring of a graph and its applications in computing the distinguishing chromatic number of disconnected graphs. We introduce two families of uniquely distinguishing colorable graphs, namely type 1 and type 2, and show that every disconnected uniquely distinguishing colorable graph is the union of two isomorphic graphs of type 2. We obtain some results on bipartite uniquely distinguishing colorable graphs and show that any uniquely distinguishing $n$-colorable tree with $ n \geq 3$ is a star graph. For a connected graph $G$, we prove that $\chi_D(G\cup G)=\chi_D(G)+1$ if and only if $G$ is uniquely distinguishing colorable of type 1. Also, a characterization of all graphs $G$ of order $n$ with the property that $\chi_{D}(G\cup G) = \chi_{D}(G) = k$, where $k=n-2, n-1, n$, is given in this paper. Moreover, we determine all graphs $G$ of order $n$ with the property that $\chi_{D}(G\cup G) = \chi_{D}(G)+1 = \ell$, where $\ell=n-1, n, n+1$. Finally, we investigate the family of connected graphs $G$ with $\chi_{D}(G\cup G) = \chi_{D}(G)+1 = 3$.

##### 6.Transitive path decompositions of Cartesian products of complete graphs

**Authors:**Ajani De Vas Gunasekara, Alice Devillers

**Abstract:** An $H$-decomposition of a graph $\Gamma$ is a partition of its edge set into subgraphs isomorphic to $H$. A transitive decomposition is a special kind of $H$-decomposition that is highly symmetrical in the sense that the subgraphs (copies of $H$) are preserved and transitively permuted by a group of automorphisms of $\Gamma$. This paper concerns transitive $H$-decompositions of the graph $K_n \Box K_n$ where $H$ is a path. When $n$ is an odd prime, we present a construction for a transitive path decomposition where the paths in the decomposition are arbitrary large.

##### 7.Polynomization of the Bessenrodt-Ono type inequalities for A-partition functions

**Authors:**Krystian Gajdzica, Bernhard Heim, Markus Neuhauser

**Abstract:** For an arbitrary set or multiset $A$ of positive integers, we associate the $A$-partition function $p_A(n)$ (that is the number of partitions of $n$ whose parts belong to $A$). We also consider the analogue of the $k$-colored partition function, namely, $p_{A,-k}(n)$. Further, we define a family of polynomials $f_{A,n}(x)$ which satisfy the equality $f_{A,n}(k)=p_{A,-k}(n)$ for all $n\in\mathbb{Z}_{\geq0}$ and $k\in\mathbb{N}$. This paper concerns the polynomization of the Bessenrodt--Ono type inequality for $f_{A,n}(x)$: \begin{align*} f_{A,a}(x)f_{A,b}(x)>f_{A,a+b}(x), \end{align*} where $a$ and $b$ are arbitrary positive integers; and delivers some efficient criteria for its solutions. Moreover, we also investigate a few basic properties related to both functions $f_{A,n}(x)$ and $f_{A,n}'(x)$.

##### 8.New constructions of non-regular cospectral graphs

**Authors:**Suliman Hamud, Abraham Berman

**Abstract:** We consider two types of joins of graphs $G_{1}$ and $G_{2}$, $G_{1}\veebar G_{2}$ - the Neighbors Splitting Join and $G_{1}\underset{=}{\lor}G_{2}$ - the Non Neighbors Splitting Join, and compute the adjacency characteristic polynomial, the Laplacian characteristic polynomial and the signless Laplacian characteristic polynomial of these joins. When $G_{1}$ and $G_{2}$ are regular, we compute the adjacency spectrum, the Laplacian spectrum, the signless Laplacian spectrum of $G_{1}\underset{=}{\lor}G_{2}$ and the normalized Laplacian spectrum of $G_{1}\veebar G_{2}$ and $G_{1}\underset{=}{\lor}G_{2}$. We use these results to construct non regular, non isomorphic graphs that are cospectral with respect to the four matrices: adjacency, Laplacian , signless Laplacian and normalized Laplacian.

##### 9.An imperceptible connection between the Clebsch--Gordan coefficients of $U_q(\mathfrak{sl}_2)$ and the Terwilliger algebras of Grassmann graphs

**Authors:**Hau-Wen Huang

**Abstract:** The Clebsch--Gordan coefficients of $U(\mathfrak{sl}_2)$ are expressible in terms of Hahn polynomials. The phenomenon can be explained by an algebra homomorphism from the universal Hahn algebra $\mathcal H$ into $U(\mathfrak{sl}_2)\otimes U(\mathfrak{sl}_2)$. Let $\Omega$ denote a finite set and $2^\Omega$ denote the power set of $\Omega$. It is generally known that $\mathbb C^{2^\Omega}$ supports a $U(\mathfrak{sl}_2)$-module. Fix an element $x_0\in 2^\Omega$. By the linear isomorphism $\mathbb C^{2^\Omega}\to \mathbb C^{2^{\Omega\setminus x_0}}\otimes \mathbb C^{2^{x_0}}$ given by $x\mapsto (x\setminus x_0)\otimes (x\cap x_0)$ for all $x\in 2^\Omega$, this induces a $U(\mathfrak{sl}_2)\otimes U(\mathfrak{sl}_2)$-module structure on $\mathbb C^{2^\Omega}$. Pulling back via the algebra homomorphism $\mathcal H\to U(\mathfrak{sl}_2)\otimes U(\mathfrak{sl}_2)$, the $U(\mathfrak{sl}_2)\otimes U(\mathfrak{sl}_2)$-module $\mathbb C^{2^\Omega}$ forms an $\mathcal H$-module. The $\mathcal H$-module $\mathbb C^{2^\Omega}$ enfolds the Terwilliger algebra of a Johnson graph. This result connects these two seemingly irrelevant topics: The Clebsch--Gordan coefficients of $U(\mathfrak{sl}_2)$ and the Terwilliger algebras of Johnson graphs. Unfortunately some steps break down in the $q$-analog case. By making detours, the imperceptible connection between the Clebsch--Gordan coefficients of $U_q(\mathfrak{sl}_2)$ and the Terwilliger algebras of Grassmann graphs is successfully disclosed in this paper.

##### 10.Grand Motzkin paths and $\{0,1,2\}$-trees -- a simple bijection

**Authors:**Helmut Prodinger

**Abstract:** A well-known bijection between Motzkin paths and ordered trees with outdegree always $\le2$, is lifted to Grand Motzkin paths (the nonnegativity is dropped) and an ordered list of an odd number of such $\{0,1,2\}$ trees. This offers an alternative to a recent paper by Rocha and Pereira Spreafico.

##### 1.A new definition of upward planar order

**Authors:**Xuexing Lu

**Abstract:** We give a more coherent and apparent definition of upward planar order.

##### 2.Powers of planar graphs, product structure, and blocking partitions

**Authors:**Marc Distel, Robert Hickingbotham, Michał T. Seweryn, David R. Wood

**Abstract:** We prove that the $k$-power of any planar graph $G$ is contained in $H\boxtimes P\boxtimes K_{f(\Delta(G),k)}$ for some graph $H$ with treewidth $15\,288\,899$, some path $P$, and some function $f$. This resolves an open problem of Ossona de Mendez. In fact, we prove a more general result in terms of shallow minors that implies similar results for many `beyond planar' graph classes, without dependence on $\Delta(G)$. For example, we prove that every $k$-planar graph is contained in $H\boxtimes P\boxtimes K_{f(k)}$ for some graph $H$ with treewidth $15\,288\,899$ and some path $P$, and some function $f$. This resolves an open problem of Dujmovi\'c, Morin and Wood. We generalise all these results for graphs of bounded Euler genus, still with an absolute bound on the treewidth. At the heart of our proof is the following new concept of independent interest. An $\ell$-blocking partition of a graph $G$ is a partition of $V(G)$ into connected sets such that every path of length greater than $\ell$ in $G$ contains at least two vertices in one part. We prove that every graph of Euler genus $g$ has a $894$-blocking partition with parts of size bounded by a function of $\Delta(G)$ and $g$. Motivated by this result, we study blocking partitions in their own right. We show that every graph has a $2$-blocking partition with parts of size bounded by a function of $\Delta(G)$ and $\textrm{tw}(G)$. On the other hand, we show that 4-regular graphs do not have $\ell$-blocking partitions with bounded size parts.

##### 3.Budget-constrained cut problems

**Authors:**Justo Puerto, José L. Sainz-Pardo

**Abstract:** The minimum and maximum cuts of an undirected edge-weighted graph are classic problems in graph theory. While the Min-Cut Problem can be solved in P, the Max-Cut Problem is NP-Complete. Exact and heuristic methods have been developed for solving them. For both problems, we introduce a natural extension in which cutting an edge induces a cost. Our goal is to find a cut that minimizes the sum of the cut weights but, at the same time, restricts its total cut cost to a given budget. We prove that both restricted problems are NPComplete and we also study some of its properties. Finally, we develop exact algorithms to solve both as well as a non-exact algorithm for the min-cut case based on a Lagreangean relaxation that generally provides optimal solutions. Their performance is reported by an extensive computational experience.

##### 4.Small sunflowers and the structure of slice rank decompositions

**Authors:**Thomas Karam

**Abstract:** Let $d \ge 3$ be an integer. We show that whenever an order-$d$ tensor admits $d+1$ decompositions according to Tao's slice rank, if the linear subspaces spanned by their one-variable functions constitute a sunflower for each choice of special coordinate, then the tensor admits a decomposition where these linear subspaces are contained in the centers of these respective sunflowers. As an application, we deduce that for every nonnegative integer $k$ and every finite field $\mathbb{F}$ there exists an integer $C(d,k,|\mathbb{F}|)$ such that every order-$d$ tensor with slice rank $k$ over $\mathbb{F}$ admits at most $C(d,k,|\mathbb{F}|)$ decompositions with length $k$, up to a class of transformations that can be easily described.

##### 5.Fan's lemma via bistellar moves

**Authors:**Tomáš Kaiser, Matěj Stehlík

**Abstract:** Pachner proved that all closed combinatorially equivalent combinatorial manifolds can be transformed into each other by a finite sequence of bistellar moves. We prove an analogue of Pachner's theorem for combinatorial manifolds with a free Z2-action, and use it to give a combinatorial proof of Fan's lemma about labellings of centrally symmetric triangulations of spheres. Similarly to other combinatorial proofs, we must assume an additional property of the triangulation for the proof to work. However, unlike the other combinatorial proofs, no such assumption is needed for dimensions at most 3.

##### 6.Star-critical Ramsey numbers and regular Ramsey numbers for stars

**Authors:**Zhidan Luo

**Abstract:** Let $G$ be a graph, $H$ be a subgraph of $G$, and let $G- H$ be the graph obtained from $G$ by removing a copy of $H$. Let $K_{1, n}$ be the star on $n+ 1$ vertices. Let $t\geq 2$ be an integer and $H_{1}, \dots, H_{t}$ and $H$ be graphs, and let $H\rightarrow (H_{1}, \dots, H_{t})$ denote that every $t$ coloring of $E(H)$ yields a monochromatic copy of $H_{i}$ in color $i$ for some $i\in [t]$. Ramsey number $r(H_{1}, \dots, H_{t})$ is the minimum integer $N$ such that $K_{N}\rightarrow (H_{1}, \dots, H_{t})$. Star-critical Ramsey number $r_{*}(H_{1}, \dots, H_{t})$ is the minimum integer $k$ such that $K_{N}- K_{1, N- 1- k}\rightarrow (H_{1}, \dots, H_{t})$ where $N= r(H_{1}, \dots, H_{t})$. Let $rr(H_{1}, \dots, H_{t})$ be the regular Ramsey number for $H_{1}, \dots, H_{t}$, which is the minimum integer $r$ such that if $G$ is an $r$-regular graph on $r(H_{1}, \dots, H_{t})$ vertices, then $G\rightarrow (H_{1}, \dots, H_{t})$. Let $m_{1}, \dots, m_{t}$ be integers larger than one, exactly $k$ of which are even. In this paper, we prove that if $k\geq 2$ is even, then $r_{*}(K_{1, m_{1}}, \dots, K_{1, m_{t}})= \sum_{i= 1}^{t} m_{i}- t+ 1- \frac{k}{2}$ which disproves a conjecture of Budden and DeJonge in 2022. Furthermore, we prove that if $k\geq 2$ is even, then $rr(K_{1, m_{1}}, \dots, K_{1, m_{t}})= \sum_{i= 1}^{t} m_{i}- t$. Otherwise, $rr(K_{1, m_{1}}, \dots, K_{1, m_{t}})= \sum_{i= 1}^{t} m_{i}- t+ 1$.

##### 7.Counting spanning subgraphs in dense hypergraphs

**Authors:**Richard Montgomery, Matías Pavez-Signé

**Abstract:** We give a simple method to estimate the number of distinct copies of some classes of spanning subgraphs in hypergraphs with high minimum degree. In particular, for each $k\geq 2$ and $1\leq \ell\leq k-1$, we show that every $k$-graph on $n$ vertices with minimum codegree at least $$\cases{\left(\dfrac{1}{2}+o(1)\right)n & if $(k-\ell)\mid k$,\\ & \\ \left(\dfrac{1}{\lceil \frac{k}{k-\ell}\rceil(k-\ell)}+o(1)\right)n & if $(k-\ell)\nmid k$,}$$ contains $\exp(n\log n-\Theta(n))$ Hamilton $\ell$-cycles as long as $(k-\ell)\mid n$. When $(k-\ell)\mid k$ this gives a simple proof of a result of Glock, Gould, Joos, K\"uhn and Osthus, while, when $(k-\ell)\nmid k$ this gives a weaker count than that given by Ferber, Hardiman and Mond or, when $\ell<k/2$, by Ferber, Krivelevich and Sudakov, but one that holds for an asymptotically optimal minimum codegree bound.

##### 8.Packing $T$-connectors in graphs needs more connectivity

**Authors:**Roman Čada, Adam Kabela, Tomáš Kaiser, Petr Vrána

**Abstract:** Strengthening the classical concept of Steiner trees, West and Wu [J. Combin. Theory Ser. B 102 (2012), 186--205] introduced the notion of a $T$-connector in a graph $G$ with a set $T$ of terminals. They conjectured that if the set $T$ is $3k$-edge-connected in $G$, then $G$ contains $k$ edge-disjoint $T$-connectors. We disprove this conjecture by constructing infinitely many counterexamples for $k=1$ and for each even $k$.

##### 9.The Cactus Group Property for Ordinal Sums of Disjoint Unions of Chains

**Authors:**Son Nguyen

**Abstract:** We study the action of Bender-Knuth involutions on linear extensions of posets and identify LE-cactus posets, i.e. those for which the cactus relations hold. It was conjectured in \cite{chiang2023bender} that d-complete posets are LE-cactus. Among the non-d-complete posets that are LE-cactus, one notable family is ordinal sums of antichains. In this paper, we characterize the LE-cactus posets in a more general family, namely ordinal sums of disjoint unions of chains.

##### 10.Web invariants for flamingo Specht modules

**Authors:**Chris Fraser, Rebecca Patrias, Oliver Pechenik, Jessica Striker

**Abstract:** Webs yield an especially important realization of certain Specht modules, irreducible representations of symmetric groups, as they provide a pictorial basis with a convenient diagrammatic calculus. In recent work, the last three authors associated polynomials to noncrossing partitions without singleton blocks, so that the corresponding polynomials form a web basis of the pennant Specht module $S^{(d,d,1^{n-2d})}$. These polynomials were interpreted as global sections of a line bundle on a 2-step partial flag variety. Here, we both simplify and extend this construction. On the one hand, we show that these polynomials can alternatively be situated in the homogeneous coordinate ring of a Grassmannian, instead of a 2-step partial flag variety, and can be realized as tensor invariants of classical (but highly nonplanar) tensor diagrams. On the other hand, we extend these ideas from the pennant Specht module $S^{(d,d,1^{n-2d})}$ to more general flamingo Specht modules $S^{(d^r,1^{n-rd})}$. In the hook case $r=1$, we obtain a spanning set that can be restricted to a basis in various ways. In the case $r>2$, we obtain a basis of a well-behaved subspace of $S^{(d^r,1^{n-rd})}$, but not of the entire module.

##### 11.Local antimagic chromatic number of partite graphs

**Authors:**C. R. Pavithra, A. V. Prajeesh, V. S. Sarath

**Abstract:** Let $G$ be a connected graph with $|V| = n$ and $|E| = m$. A bijection $f:E\rightarrow \{1,2,...,m\}$ is called a local antimagic labeling of $G$ if for any two adjacent vertices $u$ and $v$, $w(u) \neq w(v)$, where $w(u) = \sum_{e \in E(u)}f(e)$, and $E(u)$ is the set of edges incident to $u$. Thus, any local antimagic labeling induces a proper vertex coloring of $G$ where the vertex $v$ is assigned the color $w(v)$. The local antimagic chromatic number is the minimum number of colors taken over all colorings induced by local antimagic labelings of $G$. Let $m,n > 1$. In this paper, the local antimagic chromatic number of a complete tripartite graph $K_{1,m,n}$, and $r$ copies of a complete bipartite graph $K_{m,n}$ where $m \not \equiv n \bmod 2$ are determined.

##### 1.Invariants of Quadratic Forms and applications in Design Theory

**Authors:**Oliver W. Gnilke, Padraig O Cathain, Oktay Olmez, Guillermo Nunez Ponasso

**Abstract:** The study of regular incidence structures such as projective planes and symmetric block designs is a well established topic in discrete mathematics. Work of Bruck, Ryser and Chowla in the mid-twentieth century applied the Hasse-Minkowski local-global theory for quadratic forms to derive non-existence results for certain design parameters. Several combinatorialists have provided alternative proofs of this result, replacing conceptual arguments with algorithmic ones. In this paper, we show that the methods required are purely linear-algebraic in nature and are no more difficult conceptually than the theory of the Jordan Canonical Form. Computationally, they are rather easier. We conclude with some classical and recent applications to design theory, including a novel application to the decomposition of incidence matrices of symmetric designs.

##### 2.Algebraic connectivity of Kronecker products of line graphs

**Authors:**Shivani Chauhan, A. Satyanarayana Reddy

**Abstract:** Let $X$ be a tree with $n$ vertices and $L(X)$ be its line graph. In this work, we completely characterize the trees for which the algebraic connectivity of $L(X)\times K_m$ is equal to $m-1$, where $\times$ denotes the Kronecker product. We provide a few necessary and sufficient conditions for $L(X)\times K_m$ to be Laplacian integral. The algebraic connectivity of $L(X)\times K_m$, where $X$ is a tree of diameter $4$ and $k$-book graph is discussed.

##### 3.The next case of Andrásfai's conjecture

**Authors:**Tomasz Łuczak, Joanna Polcyn, Christian Reiher

**Abstract:** Let $\mathrm{ex}(n,s)$ denote the maximum number of edges in a triangle-free graph on $n$ vertices which contains no independent sets larger than $s$. The behaviour of $\mathrm{ex}(n,s)$ was first studied by Andr\'asfai, who conjectured that for $s>n/3$ this function is determined by appropriately chosen blow-ups of so called Andr\'asfai graphs. Moreover, he proved $\mathrm{ex}(n, s)=n^2-4ns+5s^2$ for $s/n\in [2/5, 1/2]$ and in earlier work we obtained $\mathrm{ex}(n, s)=3n^2-15ns+20s^2$ for $s/n\in [3/8, 2/5]$. Here we make the next step in the quest to settle Andr\'asfai's conjecture by proving $\mathrm{ex}(n, s)=6n^2-32ns+44s^2$ for $s/n\in [4/11, 3/8]$.

##### 4.On maximal cliques in the graph of simplex codes

**Authors:**Mariusz Kwiatkowski, Mark Pankov

**Abstract:** The induced subgraph of the corresponding Grassmann graph formed by simplex codes is considered. We show that this graph, as the Grassmann graph, contains two types of maximal cliques. For any two cliques of the first type there is a monomial linear automorphism transferring one of them to the other. Cliques of the second type are more complicated and can contain different numbers of elements.

##### 5.PED and POD partitions: combinatorial proofs of recurrence relations

**Authors:**Cristina Ballantine, Amanda Welch

**Abstract:** PED partitions are partitions with even parts distinct while odd parts are unrestricted. Similarly, POD partitions have distinct odd parts while even parts are unrestricted. Merca proved several recurrence relations analytically for the number of PED partitions of $n$. They are similar to the recurrence relation for the number of partitions of $n$ given by Euler's pentagonal number theorem. We provide combinatorial proofs for all of these theorems and also for the pentagonal number theorem for PED partitions proved analytically by Fink, Guy, and Krusemeyer. Moreover, we prove combinatorially a recurrence for POD partitions given by Ballantine and Merca, Beck-type identities involving PED and POD partitions, and several other results about PED and POD partitions.

##### 6.Generalizations of POD and PED partitions

**Authors:**Cristina Ballantine, Amanda Welch

**Abstract:** Partitions with even (respectively odd) parts distinct and all other parts unrestricted are often referred to as PED (respectively POD) partitions. In this article, we generalize these notions and study sets of partitions in which parts with fixed residue(s) modulo r are distinct while all other parts are unrestricted. We also study partitions in which parts divisible by r (respectively congruent to r modulo 2r) must occur with multiplicity greater than one.

##### 7.Shared ancestry graphs and symbolic arboreal maps

**Authors:**Katharina T. Huber, Vincent Moulton, Guillaume E. Scholz

**Abstract:** A network $N$ on a finite set $X$, $|X|\geq 2$, is a connected directed acyclic graph with leaf set $X$ in which every root in $N$ has outdegree at least 2 and no vertex in $N$ has indegree and outdegree equal to 1; $N$ is arboreal if the underlying unrooted, undirected graph of $N$ is a tree. Networks are of interest in evolutionary biology since they are used, for example, to represent the evolutionary history of a set $X$ of species whose ancestors have exchanged genes in the past. For $M$ some arbitrary set of symbols, $d:{X \choose 2} \to M \cup \{\odot\}$ is a symbolic arboreal map if there exists some arboreal network $N$ whose vertices with outdegree two or more are labelled by elements in $M$ and so that $d(\{x,y\})$, $\{x,y\} \in {X \choose 2}$, is equal to the label of the least common ancestor of $x$ and $y$ in $N$ if this exists and $\odot$ else. Important examples of symbolic arboreal maps include the symbolic ultrametrics, which arise in areas such as game theory, phylogenetics and cograph theory. In this paper we show that a map $d:{X \choose 2} \to M \cup \{\odot\}$ is a symbolic arboreal map if and only if $d$ satisfies certain 3- and 4-point conditions and the graph with vertex set $X$ and edge set consisting of those pairs $\{x,y\} \in {X \choose 2}$ with $d(\{x,y\}) \neq \odot$ is Ptolemaic. To do this, we introduce and prove a key theorem concerning the shared ancestry graph for a network $N$ on $X$, where this is the graph with vertex set $X$ and edge set consisting of those $\{x,y\} \in {X \choose 2}$ such that $x$ and $y$ share a common ancestor in $N$. In particular, we show that for any connected graph $G$ with vertex set $X$ and edge clique cover $K$ in which there are no two distinct sets in $K$ with one a subset of the other, there is some network with $|K|$ roots and leaf set $X$ whose shared ancestry graph is $G$.

##### 8.Demazure weaves for reduced plabic graphs (with a proof that Muller-Speyer twist is Donaldson-Thomas)

**Authors:**Roger Casals, Ian Le, Melissa Sherman-Bennett, Daping Weng

**Abstract:** First, this article develops the theory of weaves and their cluster structures for the affine cones of positroid varieties. In particular, we explain how to construct a weave from a reduced plabic graph, show it is Demazure, compare their associated cluster structures, and prove that the conjugate surface of the graph is Hamiltonian isotopic to the Lagrangian filling associated to the weave. The T-duality map for plabic graphs has a surprising key role in the construction of these weaves. Second, we use the above established bridge between weaves and reduced plabic graphs to show that the Muller-Speyer twist map on positroid varieties is the Donaldson-Thomas transformation. This latter statement implies that the Muller-Speyer twist is a quasi-cluster automorphism. An additional corollary of our results is that target labeled seeds and the source labeled seeds are related by a quasi-cluster transformation.

##### 9.Tropicalizing the Graph Profile of Some Almost-Stars

**Authors:**Maria Dascălu, Annie Raymond

**Abstract:** Many important problems in extremal combinatorics can be stated as certifying polynomial inequalities in graph homomorphism numbers, and in particular, many ask to certify pure binomial inequalities. For a fixed collection of graphs $\mathcal{U}$, the tropicalization of the graph profile of $\mathcal{U}$ essentially records all valid pure binomial inequalities involving graph homomorphism numbers for graphs in $\mathcal{U}$. Building upon ideas and techniques described by Blekherman and Raymond in 2022, we compute the tropicalization of the graph profile for $K_1$ and $S_{2,1^k}$-trees, almost-star graphs with one branch containing two edges and $k$ branches containing one edge. This allows pure binomial inequalities in homomorphism numbers (or densities) for these graphs to be verified through an explicit linear program where the number of variables is equal to the number of edges in the biggest $S_{2,1^k}$-tree involved.

##### 1.A note on Hadwiger's conjecture: Another proof that every 4-chromatic graph has a $K_4$ minor

**Authors:**Daniel Cooper McDonald

**Abstract:** The first non-obvious case of Hadwiger's Conjecture states that every graph $G$ with chromatic number at least 4 has a $K_4$ minor. We give a new proof that derives the $K_4$ minor from a proper 3-coloring of a subgraph of $G$.

##### 2.Galois points for a finite graph

**Authors:**Satoru Fukasawa, Tsuyoshi Miezaki

**Abstract:** This paper introduces the notion of a Galois point for a finite graph, using the theory of linear systems of divisors for graphs discovered by Baker and Norine. We present a new characterization of complete graphs in terms of Galois points.

##### 3.Bulgarian Solitaire: A new representation for depth generating functions

**Authors:**A. J. Harris, Son Nguyen

**Abstract:** Bulgarian Solitaire is an interesting self-map on the set of integer partitions of a fixed number $n$. As a finite dynamical system, its long-term behavior is well-understood, having recurrent orbits parametrized by necklaces of beads with two colors black $B$ and white $W$. However, the behavior of the transient elements within each orbit is much less understood. Recent work of Pham considered the orbits corresponding to a family of necklaces $P^\ell$ that are concatenations of $\ell$ copies of a fixed primitive necklace $P$. She proved striking limiting behavior as $\ell$ goes to infinity: the level statistic for the orbit, counting how many steps it takes a partition to reach the recurrent cycle, has a limiting distribution, whose generating function $H_p(x)$ is rational. Pham also conjectured that $H_P(x), H_{P^*}(x)$ share the same denominator whenever $P^*$ is obtained from $P$ by reading it backwards and swapping $B$ for $W$. Here we introduce a new representation of Bulgarian Solitaire that is convenient for the study of these generating functions. We then use it to prove two instances of Pham's conjecture, showing that $$H_{BWBWB \cdots WB}(x)=H_{WBWBW \cdots BW}(x)$$ and that $H_{BWWW\cdots W}(x),H_{WBBB\cdots B}(x)$ share the same denominator.

##### 4.Optimal chromatic bound for ($P_3\cup P_2$, house)-free graphs

**Authors:**Rui Li, Di Wu, Jinfeng Li

**Abstract:** Let $G$ and $H$ be two vertex disjoint graphs. The {\em union} $G\cup H$ is the graph with $V(G\cup H)=V(G)\cup V(H)$ and $E(G\cup H)=E(G)\cup E(H)$. We use $P_k$ to denote a {\em path} on $k$ vertices, use {\em house} to denote the complement of $P_5$. In this paper, we show that $\chi(G)\le2\omega(G)$ if $G$ is ($P_3\cup P_2$, house)-free. Moreover, this bound is optimal when $\omega(G)\ge2$.

##### 5.Extensions of transversal valuated matroids

**Authors:**Alex Fink, Jorge Alberto Olarte

**Abstract:** Following up on our previous work, we study single-element extensions of transversal valuated matroids. We show that tropical presentations of valuated matroids with a minimal set of finite entries enjoy counterparts of the properties proved by Bonin and de Mier of minimal non-valuated transversal presentations.

##### 6.Erdős-Gyárfás Conjecture for $P_{10}$-free Graphs

**Authors:**Zhiquan Hu, Changlong Shen

**Abstract:** Let $P_{10}$ be a path on $10$ vertices. A graph is said to be $P_{10}$-free if it does not contain $P_{10}$ as an induced subgraph. The well-known Erd\H{o}s-Gy\'{a}rf\'{a}s Conjecture states that every graph with minimum degree at least three has a cycle whose length is a power of $2$. In this paper, we show that every $P_{10}$-free graph with minimum degree at least three contains a cycle of length $4$ or $8$. This implies that the conjecture is true for $P_{10}$-free graphs.

##### 7.Towards the Automorphism Conjecture I: Combinatorial Control and Compensation for Factorials

**Authors:**Bernd S. W. Schröder

**Abstract:** This paper exploits adjacencies between the orbits of an ordered set P and a consequence of the classification of finite simple groups to, in many cases, exponentially bound the number of automorphisms. Results clearly identify the structures which currently prevent the proof of such an exponential bound, or which indeed inflate the number of automorphisms beyond such a bound. This is a first step towards a possible resolution of the Automorphism Conjecture for ordered sets.

##### 1.Note on disjoint faces in simple topological graphs

**Authors:**Ji Zeng

**Abstract:** We prove that every $n$-vertex complete simple topological graph generates at least $\Omega(n)$ pairwise disjoint $4$-faces. This improves upon a recent result by Hubard and Suk. As an immediate corollary, every $n$-vertex complete simple topological graph drawn in the unit square generates a $4$-face with area at most $O(1/n)$. This can be seen as a topological variant of Heilbronn's problem for $4$-faces. We construct examples showing that our result is asymptotically tight. We also discuss the similar problem for $k$-faces with arbitrary $k\geq 3$.

##### 2.Hurwitz numbers for reflection groups III: Uniform formulas

**Authors:**Theo Douvropoulos, Joel Brewster Lewis, Alejandro H. Morales

**Abstract:** We give uniform formulas for the number of full reflection factorizations of a parabolic quasi-Coxeter element in a Weyl group or complex reflection group, generalizing the formula for the genus-0 Hurwitz numbers. This paper is the culmination of a series of three.

##### 3.Pancyclic property of Paley graph

**Authors:**Yusaku Nishimura

**Abstract:** Let $G$ be an undirected graph of order $n$ and let $C_i$ be an $i$-cycle graph. $G$ is called pancyclic if $G$ contains a $C_i$ for any $i\in \{3,4,\ldots,n\}$. The Paley graph of order $q$ is a graph whose vertex set is the finite field $\mathbb{F}_q$, where $q\equiv 1\pmod{4}$. In this graph, two vertices, $a$ and $b$, are adjacent if and only if $a-b$ is nonzero square in $\mathbb{F}_q$. We prove that Paley graph of order $q$ is pancyclic for any $q \neq 5$.

##### 4.Inversion and Cubic Vectors for Permutrees

**Authors:**Daniel Tamayo Jiménez

**Abstract:** We introduce two generalizations of bracket vectors from binary trees to permutrees. These new vectors help describe algebraic and geometric properties of the rotation lattice of permutrees defined by Pilaud and Pons. The first generalization serves the role of an inversion vector for permutrees allowing us to define an explicit meet operation and provide a new constructive proof of the lattice property for permutree rotation lattices. The second generalization, which we call cubic vectors, allows for the construction of a cubic realization of these lattices which is proven to form a cubical embedding of the corresponding permutreehedra. These results specialize to those known about permutahedra and associahedra.

##### 5.Total positivity from a kind of lattice paths

**Authors:**Yu-Jie Cui, Bao-Xuan Zhu

**Abstract:** Total positivity of matrices is deeply studied and plays an important role in various branches of mathematics. The main purpose of this paper is to study total positivity of a matrix $M=[M_{n,k}]_{n,k}$ generated by the weighted lattice paths in $\mathbb{N}^2$ from the origin $(0,0)$ to the point $(k,n)$ consisting of types of steps: $(0,1)$ and $(1,t+i)$ for $0\leq i\leq \ell$, where each step $(0,1)$ from height~$n-1$ gets the weight~$b_n(\textbf{y})$ and each step $(1,t+i)$ from height~$n-t-i$ gets the weight $a_n^{(i)}(\textbf{x})$. Using an algebraic method, we prove that the $\textbf{x}$-total positivity of the weight matrix $[a_i^{(i-j)}(\textbf{x})]_{i,j}$ implies that of $M$. Furthermore, using the Lindstr\"{o}m-Gessel-Viennot lemma, we obtain that both $M$ and the Toeplitz matrix of each row sequence of $M$ with $t\geq1$ are $\textbf{x}$-totally positive under the following three cases respectively: (1) $\ell=1$, (2) $\ell=2$ and restrictions for $a_n^{(i)}$, (3) general $\ell$ and both $a^{(i)}_n$ and $b_n$ are independent of $n$. In addition, for the case (3), we show that the matrix $M$ is a Riordan array, present its explicit formula and prove total positivity of the Toeplitz matrix of the each column of $M$. In particular, from the results for Toeplitz-total positivity, we also obtain the P\'olya frequency and log-concavity of the corresponding sequence. Finally, as applications, we in a unified manner establish total positivity and the Toeplitz-total positivity for many well-known combinatorial triangles, including the Pascal triangle, the Pascal square, the Delannoy triangle, the Delannoy square, the signless Stirling triangle of the first kind, the Legendre-Stirling triangle of the first kind, the Jacobi-Stirling triangle of the first kind, the Brenti's recursive matrix, and so on.

##### 6.Ordering Candidates via Vantage Points

**Authors:**Noga Alon, Colin Defant, Noah Kravitz, Daniel G. Zhu

**Abstract:** Given an $n$-element set $C\subseteq\mathbb{R}^d$ and a (sufficiently generic) $k$-element multiset $V\subseteq\mathbb{R}^d$, we can order the points in $C$ by ranking each point $c\in C$ according to the sum of the distances from $c$ to the points of $V$. Let $\Psi_k(C)$ denote the set of orderings of $C$ that can be obtained in this manner as $V$ varies, and let $\psi^{\mathrm{max}}_{d,k}(n)$ be the maximum of $\lvert\Psi_k(C)\rvert$ as $C$ ranges over all $n$-element subsets of $\mathbb{R}^d$. We prove that $\psi^{\mathrm{max}}_{d,k}(n)=\Theta_{d,k}(n^{2dk})$ when $d \geq 2$ and that $\psi^{\mathrm{max}}_{1,k}(n)=\Theta_k(n^{4\lceil k/2\rceil -1})$. As a step toward proving this result, we establish a bound on the number of sign patterns determined by a collection of functions that are sums of radicals of nonnegative polynomials; this can be understood as an analogue of a classical theorem of Warren. We also prove several results about the set $\Psi(C)=\bigcup_{k\geq 1}\Psi_k(C)$; this includes an exact description of $\Psi(C)$ when $d=1$ and when $C$ is the set of vertices of a vertex-transitive polytope.

##### 7.An asymptotic property of quaternary additive codes

**Authors:**Jürgen Bierbrauer, Stefano Marcugini, Fernanda Pambianco

**Abstract:** Let $n_k(s)$ be the maximal length $n$ such that a quaternary additive $[n,k,n-s]_4$-code exists. We solve a natural asymptotic problem by determining the limsup $\lambda_k$ of $n_k(s)/s,$ and the smallest value of $s$ such that $n_k(s)/s=\lambda_k.$ The proof uses geometric language and is elementary.

##### 1.The Widths of Strict Outerconfluent Graphs

**Authors:**David Eppstein

**Abstract:** Strict outerconfluent drawing is a style of graph drawing in which vertices are drawn on the boundary of a disk, adjacencies are indicated by the existence of smooth curves through a system of tracks within the disk, and no two adjacent vertices are connected by more than one of these smooth tracks. We investigate graph width parameters on the graphs that have drawings in this style. We prove that the clique-width of these graphs is unbounded, but their twin-width is bounded.

##### 2.A class of trees determined by their chromatic symmetric functions

**Authors:**Yuzhenni Wang, Xingxing Yu, Xiao-Dong Zhang

**Abstract:** Stanley introduced the concept of chromatic symmetric functions of graphs which extends and refines the notion of chromatic polynomials of graphs. Stanley further conjectured that trees are determined up to isomorphism by their chromatic symmetric functions. In this paper, we study various representations of chromatic symmetric functions. We verify Stanley's conjecture for the class of trees with exactly two vertices of degree at least 3.

##### 3.Integration on complex Grassmannians, deformed monotone Hurwitz numbers, and interlacing phenomena

**Authors:**Xavier Coulter, Norman Do, Ellena Moskovsky

**Abstract:** We introduce a family of polynomials, which arise in three distinct ways: in the large $N$ expansion of a matrix integral, as a weighted enumeration of factorisations of permutations, and via the topological recursion. More explicitly, we interpret the complex Grassmannian $\mathrm{Gr}(M,N)$ as the space of $N \times N$ idempotent Hermitian matrices of rank $M$ and develop a Weingarten calculus to integrate products of matrix elements over it. In the regime of large $N$ and fixed ratio $\frac{M}{N}$, such integrals have expansions whose coefficients count factorisations of permutations into monotone sequences of transpositions, with each sequence weighted by a monomial in $t = 1 - \frac{N}{M}$. This gives rise to the desired polynomials, which specialise to the monotone Hurwitz numbers when $t = 1$. These so-called deformed monotone Hurwitz numbers satisfy a cut-and-join recursion, a one-point recursion, and the topological recursion. Furthermore, we conjecture on the basis of overwhelming empirical evidence that the deformed monotone Hurwitz numbers are real-rooted polynomials whose roots satisfy remarkable interlacing phenomena. An outcome of our work is the viewpoint that the topological recursion can be used to "topologise" sequences of polynomials, and we claim that the resulting families of polynomials may possess interesting properties. As a further case study, we consider a weighted enumeration of dessins d'enfant and conjecture that the resulting polynomials are also real-rooted and satisfy analogous interlacing properties.

##### 4.Characterization of rings with genus two prime ideal sum graphs

**Authors:**Praveen Mathil, Jitender Kumar

**Abstract:** Let $R$ be a commutative ring with unity. The prime ideal sum graph of the ring $R$ is a simple undirected graph whose vertex set is the set of nonzero proper ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if $I + J$ is a prime ideal of $R$. In this paper, we characterize all the finite non-local commutative rings whose prime ideal sum graph is of genus $2$.

##### 1.Critical $(P_5,dart)$-Free Graphs

**Authors:**Wen Xia, Shenwei Huang

**Abstract:** Given two graphs $H_1$ and $H_2$, a graph is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ and $H_2$. Let $P_t$ be the path on $t$ vertices. A dart is the graph obtained from a diamond by adding a new vertex and making it adjacent to exactly one vertex which degree is 3 in the diamond. In this paper, we show that there are finitely many $k$-vertex-critical $(P_5,dart)$-free graphs for $k \ge 1$. To prove the results we use induction on $k$ and perform a careful structural analysis via Strong Perfect Graph Theorem combined with the pigeonhole principle based on the properties of vertex-critical graphs. Our results imply the existence of a polynomial-time certifying algorithm to decide the $k$-colorability of $(P_5,dart)$-free graphs for $k \ge 1$ where the certificate is either a $k$-coloring or a $(k+1)$-vertex-critical induced subgraph.

##### 2.Tyshkevich's Graph Decomposition and the Distinguishing Numbers of Unigraphs

**Authors:**Christine T. Cheng

**Abstract:** A $c$-labeling $\phi: V(G) \rightarrow \{1, 2, \hdots, c \}$ of graph $G$ is distinguishing if, for every non-trivial automorphism $\pi$ of $G$, there is some vertex $v$ so that $\phi(v) \neq \phi(\pi(v))$. The distinguishing number of $G$, $D(G)$, is the smallest $c$ such that $G$ has a distinguishing $c$-labeling. We consider a compact version of Tyshkevich's graph decomposition theorem where trivial components are maximally combined to form a complete graph or a graph of isolated vertices. Suppose the compact canonical decomposition of $G$ is $G_{k} \circ G_{k-1} \circ \cdots \circ G_1 \circ G_0$. We prove that $\phi$ is a distinguishing labeling of $G$ if and only if $\phi$ is a distinguishing labeling of $G_i$ when restricted to $V(G_i)$ for $i = 0, \hdots, k$. Thus, $D(G) = \max \{D(G_i), i = 0, \hdots, k \}$. We then present an algorithm that computes the distinguishing number of a unigraph in linear time.

##### 3.1-Konig-Egervary Graphs

**Authors:**Vadim E. Levit, Eugen Mandrescu

**Abstract:** Let $\alpha(G)$ denote the cardinality of a maximum independent set, while $\mu(G)$ be the size of a maximum matching in $G=\left( V,E\right) $. It is known that if $\alpha(G)+\mu(G)=\left\vert V\right\vert $, then $G$ is a \textit{K\"{o}nig-Egerv\'{a}ry graph. If $\alpha (G)+\mu(G)=\left\vert V\right\vert -1$, then $G$ is an $1$-K\"{o}nig-Egerv\'{a}ry graph. If $G$ is not a K\"{o}nig-Egerv\'{a}ry graph, and there exists a vertex $v\in V$ (an edge $e\in E$) such that $G-v$ ($G-e$) is K\"{o}nig-Egerv\'{a}ry, then $G$ is called a vertex (an edge) almost K\"{o}nig-Egerv\'{a}ry graph (respectively). In this paper, we characterize all these types of almost K\"{o}nig-Egerv\'{a}ry graphs and present interrelationships between them.

##### 4.A note on monotonicity in Maker-Breaker graph colouring games

**Authors:**Lawrence Hollom

**Abstract:** In the Maker-Breaker vertex colouring game, first publicised by Gardner in 1981, Maker and Breaker alternately colour vertices of a graph using a fixed palette, maintaining a proper colouring at all times. Maker aims to colour the whole graph, and Breaker aims to make some vertex impossible to colour. We are interested in the following question, first asked by Zhu in 1999: if Maker wins with $k$ colours available, must they also win with $k+1$? This question has remained open, attracting significant attention and being reposed for many similar games. While we cannot resolve this problem for the vertex colouring game, we can answer it in the affirmative for the game of arboricity, resolving a question of Bartnicki, Grytczuk, and Kierstead from 2008. We then consider how one might approach the question of monotonicity for the vertex colouring game, and work with a related game in which the vertices must be coloured in a prescribed order. We demonstrate that this `ordered vertex colouring game' does not have the above monotonicity property, and discuss the implications of this fact to the unordered game. Finally, we provide counterexamples to two open problems concerning a connected version of the graph colouring game.

##### 5.Nontrivial Intersecting Families in Multisets

**Authors:**Jiaqi Liao, Zequn Lv, Mengyu Cao, Mei Lu

**Abstract:** Let $ k $, $ m $ and $ n $ be three positive integers. A $ k $-multiset in $ [n]_m $ ($ m = \infty $ is allowed) is a $ k $-`set' whose elements are integers from $ \cb{1, \ldots, n} $, and each element is allowed to have at most $ m $ repetitions. A family of $ k $-multisets in $ [n]_m $ is called intersecting if every pair of $k$-multisets from the family have non-empty intersection. In this paper, we use a transpositional trick and the Hilton-Milner Theorem to give the size and structure of the largest non-trivial intersecting family of $ k $-multisets in $ [n]_m $.

##### 6.The Erdős distinct subset sums problem in a modular setting

**Authors:**Stijn Cambie, Jun Gao, Younjin Kim, Hong Liu

**Abstract:** We prove the following variant of the Erd\H{o}s distinct subset sums problem. Given $t \ge 0$ and sufficiently large $n$, every $n$-element set $A$ whose subset sums are distinct modulo $N=2^n+t$ satisfies $$\max A \ge \Big(\frac{1}{3}-o(1)\Big)N.$$ Furthermore, we provide examples showing that the constant $\frac 13$ is best possible. For small values of $t$, we characterise the structure of all sumset-distinct sets modulo $N=2^n+t$ of cardinality $n$.

##### 7.Counting two-forests and random cut size via potential theory

**Authors:**Harry Richman, Farbod Shokrieh, Chenxi Wu

**Abstract:** We prove a lower bound on the number of spanning two-forests in a graph, in terms of the number of vertices, edges, and spanning trees. This implies an upper bound on the average cut size of a random two-forest. The main tool is an identity relating the number of spanning trees and two-forests to pairwise effective resistances in a graph. Along the way, we make connections to potential theoretic invariants on metric graphs.

##### 8.Borsuk and Vázsonyi problems through Reuleaux polyhedra

**Authors:**Gyivan Lopez-Campos, Deborah Oliveros, Jorge L. Ramírez Alfonsín

**Abstract:** The Borsuk conjecture and the V\'azsonyi problem are two attractive and famous questions in discrete and combinatorial geometry, both based on the notion of diameter of a bounded sets. In this paper, we present an equivalence between the critical sets with Borsuk number 4 in $\mathbb{R}^3$ and the minimal structures for the V\'azsonyi problem by using the well-known Reuleaux polyhedra. The latter lead to a full characterization of all finite sets in $\mathbb{R}^3$ with Borsuk number 4. The proof of such equivalence needs various ingredients, in particular, we proved a conjecture dealing with strongly critical configuration for the V\'azsonyi problem and showed that the diameter graph arising from involutive polyhedra is vertex (and edge) 4-critical.

##### 1.Connectivity gaps among matroids with the same enumerative invariants

**Authors:**Joseph E. Bonin, Kevin Long

**Abstract:** Many important enumerative invariants of a matroid can be obtained from its Tutte polynomial, and many more are determined by two stronger invariants, the $\mathcal{G}$-invariant and the configuration of the matroid. We show that the same is not true of the most basic connectivity invariants. Specifically, we show that for any positive integer $n$, there are pairs of matroids that have the same configuration (and so the same $\mathcal{G}$-invariant and the same Tutte polynomial) but the difference between their Tutte connectivities exceeds $n$, and likewise for vertical connectivity and branch-width. The examples that we use to show this, which we construct using an operation that we introduce, are transversal matroids that are also positroids.

##### 2.A formula for the base size of the symmetric group in its action on subsets

**Authors:**Giovanni Mecenero, Pablo Spiga

**Abstract:** Given two positive integers $n$ and $k$, we obtain a formula for the base size of the symmetric group of degree $n$ in its action on $k$-subsets. Then, we use this formula to compute explicitly the base size for each $n$ and for each $k\le 14$.

##### 3.Fractional revival on semi-Cayley graphs over abelian groups

**Authors:**Jing Wang, Ligong Wang, Xiaogang Liu

**Abstract:** In this paper, we investigate the existence of fractional revival on semi-Cayley graphs over finite abelian groups. We give some necessary and sufficient conditions for semi-Cayley graphs over finite abelian groups admitting fractional revival. We also show that integrality is necessary for some semi-Cayley graphs admitting fractional revival. Moreover, we characterize the minimum time when semi-Cayley graphs admit fractional revival. As applications, we give examples of certain Cayley graphs over the generalized dihedral groups and generalized dicyclic groups admitting fractional revival.

##### 4.Everywhere unbalanced configurations

**Authors:**David Conlon, Jeck Lim

**Abstract:** An old problem in discrete geometry, originating with Kupitz, asks whether there is a fixed natural number $k$ such that every finite set of points in the plane has a line through at least two of its points where the number of points on either side of this line differ by at most $k$. We give a negative answer to a natural variant of this problem, showing that for every natural number $k$ there exists a finite set of points in the plane together with a pseudoline arrangement such that each pseudoline contains at least two points and there is a pseudoline through any pair of points where the number of points on either side of each pseudoline differ by at least $k$.

##### 5.Prime and polynomial distances in colourings of the plane

**Authors:**James Davies, Rose McCarty, Michał Pilipczuk

**Abstract:** We give two extensions of the recent theorem of the first author that the odd distance graph has unbounded chromatic number. The first is that for any non-constant polynomial $f$ with integer coefficients and positive leading coefficient, every finite colouring of the plane contains a monochromatic pair of distinct points whose distance is equal to $f(n)$ for some integer $n$. The second is that for every finite colouring of the plane, there is a monochromatic pair of points whose distance is a prime number.

##### 1.Algorithmic study of $d_2$-transitivity of graphs

**Authors:**Subhabrata Paul, Kamal Santra

**Abstract:** Let $G=(V, E)$ be a graph where $V$ and $E$ are the vertex and edge sets, respectively. For two disjoint subsets $A$ and $B$ of $V$, we say $A$ \emph{dominates} $B$ if every vertex of $B$ is adjacent to at least one vertex of $A$. A vertex partition $\pi = \{V_1, V_2, \ldots, V_k\}$ of $G$ is called a \emph{transitive partition} of size $k$ if $V_i$ dominates $V_j$ for all $1\leq i<j\leq k$. In this article, we initiate the study of a generalization of transitive partition, namely \emph{$d_2$-transitive partition}. For two disjoint subsets $A$ and $B$ of $V$, we say $A$ \emph{$d_2$-dominates} $B$ if, for every vertex of $B$, there exists a vertex in $A$, such that the distance between them is at most two. A vertex partition $\pi = \{V_1, V_2, \ldots, V_k\}$ of $G$ is called a \emph{$d_2$-transitive partition} of size $k$ if $V_i$ $d_2$-dominates $V_j$ for all $1\leq i<j\leq k$. The maximum integer $k$ for which the above partition exists is called \emph{$d_2$-transitivity} of $G$, and it is denoted by $Tr_{d_2}(G)$. The \textsc{Maximum $d_2$-Transitivity Problem} is to find a $d_2$-transitive partition of a given graph with the maximum number of parts. We show that this problem can be solved in linear time for the complement of bipartite graphs and bipartite chain graphs. On the negative side, we prove that the decision version of the \textsc{Maximum $d_2$-Transitivity Problem} is NP-complete for split graphs, bipartite graphs, and star-convex bipartite graphs.

##### 2.Sparse pancyclic subgraphs of random graphs

**Authors:**Yahav Alon, Michael Krivelevich

**Abstract:** It is known that the complete graph $K_n$ contains a pancyclic subgraph with $n+(1+o(1))\cdot \log _2 n$ edges, and that there is no pancyclic graph on $n$ vertices with fewer than $n+\log _2 (n-1) -1$ edges. We show that, with high probability, $G(n,p)$ contains a pancyclic subgraph with $n+(1+o(1))\log_2 n$ edges for $p \ge p^*$, where $p^*=(1+o(1))\ln n/n$, right above the threshold for pancyclicity.

##### 3.New constructions of NMDS self-dual codes

**Authors:**Dongchun Han, Hanbin Zhang

**Abstract:** Near maximum distance separable (NMDS) codes are important in finite geometry and coding theory. Self-dual codes are closely related to combinatorics, lattice theory, and have important application in cryptography. In this paper, we construct a class of $q$-ary linear codes and prove that they are either MDS or NMDS which depends on certain zero-sum condition. In the NMDS case, we provide an effective approach to construct NMDS self-dual codes which largely extend known parameters of such codes. In particular, we proved that for square $q$, almost $q/8$ NMDS self-dual $q$-ary codes can be constructed.

##### 4.Reconstruction of colours

**Authors:**Yury Demidovich, Yaroslav Panichkin, Maksim Zhukovskii

**Abstract:** Given a graph $G$, when is it possible to reconstruct with high probability a uniformly random colouring of its vertices in $r$ colours from its $k$-deck, i.e. a set of its induced (coloured) subgraphs of size $k$? In this paper, we reconstruct random colourings of lattices and random graphs. Recently, Narayanan and Yap proved that, for $d=2$, with high probability a random colouring of vertices of a $d$-dimensional $n$-lattice ($n\times n$ grid) is reconstructibe from its deck of all $k$-subgrids ($k\times k$ grids) if $k\geq\sqrt{2\log_2 n}+\frac{3}{4}$ and is not reconstructible if $k<\sqrt{2\log_2 n}-\frac{1}{4}$. We prove that the same "two-point concentration" result for the minimum size of subgrids that determine the entire colouring holds true in any dimension $d\geq 2$. We also prove that with high probability a uniformly random $r$-colouring of the vertices of the random graph $G(n,1/2)$ is reconstructible from its full $k$-deck if $k\geq 2\log_2 n+8$ and is not reconstructible if $k\leq\sqrt{2\log_2 n}$. We further show that the colour reconstruction algorithm for random graphs can be modified and used for graph reconstruction: we prove that with high probability $G(n,1/2)$ is reconstructible from its full $k$-deck if $k\geq 2\log_2 n+11$ (while it is not reconstructible with high probability if $k\leq 2\sqrt{\log_2 n}$). This significantly improves the best known upper bound for the minimum size of subgraphs in a deck that can be used to reconstruct the random graph with high probability.

##### 5.Inequalities Connecting the Annihilation and Independence Numbers

**Authors:**Ohr Kadrawi, Vadim E. Levit

**Abstract:** Given a graph $G$, the number of its vertices is represented by $n(G)$, while the number of its edges is denoted as $m(G)$. An independent set in a graph is a set of vertices where no two vertices are adjacent to each other and the size of the maximum independent set is denoted by $\alpha(G)$. A matching in a graph refers to a set of edges where no two edges share a common vertex and the maximum matching size is denoted by $\mu(G)$. If $\alpha(G) + \mu(G) = n(G)$, then the graph $G$ is called a K\"{o}nig-Egerv\'{a}ry graph. Considering a graph $G$ with a degree sequence $d_1 \leq d_2 \leq \cdots \leq d_n$, the annihilation number $a(G)$ is defined as the largest integer $k$ such that the sum of the first $k$ degrees in the sequence is less than or equal to $m(G)$ (Pepper, 2004). It is a known fact that $\alpha(G)$ is less than or equal to $a(G)$ for any graph $G$. Our goal is to estimate the difference between these two parameters. Specifically, we prove a series of inequalities, including $a(G) - \alpha(G) \leq \frac{\mu(G) - 1}{2}$ for trees, $a(G) - \alpha(G) \leq 2 + \mu(G) - 2\sqrt{1 + \mu(G)}$ for bipartite graphs and $a(G) - \alpha(G) \leq \mu(G) - 2$ for K\"{o}nig-Egerv\'{a}ry graphs. Furthermore, we demonstrate that these inequalities serve as tight upper bounds for the difference between the annihilation and independence numbers, regardless of the assigned value for $\mu(G)$.

##### 6.Identifiability of Homoscedastic Linear Structural Equation Models using Algebraic Matroids

**Authors:**Mathias Drton, Benjamin Hollering, Jun Wu

**Abstract:** We consider structural equation models (SEMs), in which every variable is a function of a subset of the other variables and a stochastic error. Each such SEM is naturally associated with a directed graph describing the relationships between variables. When the errors are homoscedastic, recent work has proposed methods for inferring the graph from observational data under the assumption that the graph is acyclic (i.e., the SEM is recursive). In this work, we study the setting of homoscedastic errors but allow the graph to be cyclic (i.e., the SEM to be non-recursive). Using an algebraic approach that compares matroids derived from the parameterizations of the models, we derive sufficient conditions for when two simple directed graphs generate different distributions generically. Based on these conditions, we exhibit subclasses of graphs that allow for directed cycles, yet are generically identifiable. We also conjecture a strengthening of our graphical criterion which can be used to distinguish many more non-complete graphs.

##### 7.Square Coloring of Planar Graphs with Maximum Degree at Most Five

**Authors:**Jiani Zou, Miaomiao Han, Hong-Jian Lai

**Abstract:** The square coloring of a graph $G$ is a stronger version of proper vertex coloring, requiring additionally that any two distinct vertices with a common neighbor need to be colored differently. In this paper we prove that every planar graph with maximum degree at most $5$ admits a square coloring using at most $17$ colors, which improves the upper bound $18$ recently presented by Hou, Jin, Miao, and Zhao [Graphs and Combinatorics, 2023].

##### 1.Homological algebra and poset versions of the Garland method

**Authors:**Eric Babson, Volkmar Welker

**Abstract:** Garland introduced a vanishing criterion for a characteristic zero cohomology group of a locally finite and locally connected simplicial complex. The criterion is based on the spectral gaps of the graph Laplacians of the links of faces and has turned out to be effective in a wide range of examples. In this note we extend the approach to include a range of non-simplicial (co)chain complexes associated to combinatorial structures we call Garland posets and elaborate further on the case of cubical complexes.

##### 2.Doubly even self-orthogonal codes from quasi-symmetric designs

**Authors:**Dean Crnković, Doris Dumičić Danilović, Ana Šumberac, Andrea Švob

**Abstract:** In this paper, we give a construction of doubly even self-orthogonal codes from quasi-symmetric designs. Further, we study orbit matrices of quasi-symmetric designs and give a construction of doubly even self-orthogonal codes from orbit matrices of quasi-symmetric designs of Blokhuis-Haemers type.

##### 3.Stability of Cayley graphs and Schur rings

**Authors:**Ademir Hujdurović, István Kovács

**Abstract:** A graph $\Gamma$ is said to be unstable if for the direct product $\Gamma \times K_2$, $Aut(\Gamma \times K_2)$ is not isomorphic to $Aut(\Gamma) \times \mathbb{Z}_2$. In this paper we show that a connected and non-bipartite Cayley graph $Cay(H,S)$ is unstable if and only if the set $S \times \{1\}$ belongs to a Schur ring over the group $H \times \mathbb{Z}_2$ having certain properties. The Schur rings with these properties are characterized if $H$ is an abelian group of odd order or a cyclic group of twice odd order. As an application, a short proof is given for the result of Witte Morris stating that every connected unstable Cayley graph on an abelian group of odd order has twins (Electron.~J.~Combin, 2021). As another application, sufficient and necessary conditions are given for a connected and non-bipartite circulant graph of order $2p^e$ to be unstable, where $p$ is an odd prime and $e \ge 1$.

##### 4.Balanced-chromatic number and Hadwiger-like conjectures

**Authors:**Andrea Jiménez, Jessica Mcdonald, Reza Naserasr, Kathryn Nurse, Daniel A. Quiroz

**Abstract:** Motivated by different characterizations of planar graphs and the 4-Color Theorem, several structural results concerning graphs of high chromatic number have been obtained. Toward strengthening some of these results, we consider the \emph{balanced chromatic number}, $\chi_b(\hat{G})$, of a signed graph $\hat{G}$. This is the minimum number of parts into which the vertices of a signed graph can be partitioned so that none of the parts induces a negative cycle. This extends the notion of the chromatic number of a graph since $\chi(G)=\chi_b(\tilde{G})$, where $\tilde{G}$ denotes the signed graph obtained from~$G$ by replacing each edge with a pair of (parallel) positive and negative edges. We introduce a signed version of Hadwiger's conjecture as follows. Conjecture: If a signed graph $\hat{G}$ has no negative loop and no $\tilde{K_t}$-minor, then its balanced chromatic number is at most $t-1$. We prove that this conjecture is, in fact, equivalent to Hadwiger's conjecture and show its relation to the Odd Hadwiger Conjecture. Motivated by these results, we also consider the relation between subdivisions and balanced chromatic number. We prove that if $(G, \sigma)$ has no negative loop and no $\tilde{K_t}$-subdivision, then it admits a balanced $\frac{79}{2}t^2$-coloring. This qualitatively generalizes a result of Kawarabayashi (2013) on totally odd subdivisions.

##### 5.Permutation and local permutation polynomial of maximum degree

**Authors:**Jaime Gutierrez, Jorge Jimenez Urroz

**Abstract:** Let $F_q$ be the finite field with $q$ elements and $F_q[x_1,\ldots, x_n]$ the ring of polynomials in $n$ variables over $F_q$. In this paper we consider permutation polynomials and local permutation polynomials over $F_q[x_1,\ldots, x_n]$, which define interesting generalizations of permutations over finite fields. We are able to construct permutation polynomials in $F_q[x_1,\ldots, x_n]$ of maximum degree $n(q-1)-1$ and local permutation polynomials in $F_q[x_1,\ldots, x_n]$ of maximum degree $n(q-2)$ when $q>3$, extending previous results.

##### 1.Random Walk Labelings of Perfect Trees and Other Graphs

**Authors:**Sela Fried, Toufik Mansour

**Abstract:** A Random walk labeling of a graph $G$ is any labeling of $G$ that could have been obtained by performing a random walk on $G$. Continuing two recent works, we calculate the number of random walk labelings of perfect trees, combs, and double combs, the torus $C_2\times C_n$, and the graph obtained by connecting three path graphs to form two cycles.

##### 2.A short note on cospectral and integral chain graphs for Seidel matrix

**Authors:**Santanu Mandal

**Abstract:** In this brief communication, we investigate the cospectral as well integral chain graphs for Seidel matrix, a key component to study the structural properties of equiangular lines in space. We derive a formula that allows to generate an infinite number of inequivalent chain graphs with identical spectrum. In addition, we obtain a family of Seidel integral chain graphs. This contrapositively answers a problem posed by Greaves ["Equiangular line systems and switching classes containing regular graphs", Linear Algebra Appl., (2018)] ("Does every Seidel matrix with precisely three distinct rational eigenvalues contain a regular graph in its switching class?"). Our observation is- "no".

##### 3.On the maximal sum of the entries of a matrix power

**Authors:**Sela Fried, Toufik Mansour

**Abstract:** Let $p_n$ be the maximal sum of the entries of $A^2$, where $A$ is a square matrix of size $n$, consisting of the numbers $1,2,\ldots,n^2$, each appearing exactly once. We prove that $m_n=\Theta(n^7)$. More precisely, we show that $n(240n^{6}+28n^{5}+364n^{4}+210n^{2}-28n+26-105((-1)^{n}+1))/840\leq p_n\leq n^{3}(n^{2}+1)(7n^{2}+5)/24$.

##### 4.Some results on 2-distance coloring of planar graphs with girth five

**Authors:**Zakir Deniz

**Abstract:** A vertex coloring of a graph $G$ is called a 2-distance coloring if any two vertices at a distance at most $2$ from each other receive different colors. Suppose that $G$ is a planar graph with girth $5$ and maximum degree $\Delta$. We prove that $G$ admits a $2$-distance $\Delta+7$ coloring, which improves the result of Dong and Lin (J. Comb. Optim. 32(2), 645-655, 2016). Moreover, we prove that $G$ admits a $2$-distance $\Delta+6$ coloring when $\Delta\geq 10$.

##### 5.A complete solution of the $k$-uniform supertrees with the eight largest $α$-spectral radii

**Authors:**Lou-Jun Yu, Wen-Huan Wang

**Abstract:** Let $\mathcal T (n, k)$ be the set of the $k$-uniform supertrees with $n$ vertices and $m$ edges, where $k\geq 3$, $n\geq 5$ and $m=\frac{n-1}{k-1}$. % Let $m$ be the number of the edges of the supertrees in $\mathcal T (n, k)$, where $m=\frac{n-1}{k-1}$. A conjecture concerning the supertrees with the fourth through the eighth largest $\alpha$-spectral radii in $\mathcal T (n, k)$ was proposed by You et al.\ (2020), where $0 \leq \alpha<1$, $k\geq 3$ and $m \geq 10$. This conjecture was partially solved for $1-\frac{1}{m-2}\leq \alpha <1$ and $m\geq 10$ by Wang et al.\ (2022). When $0\leq \alpha <1-\frac{1}{m-2}$ and $m \geq 10$, whether this conjecture is correct or not remains a problem to be further solved. By using a new $\rho_{\alpha}$-normal labeling method proposed in this article for computing the $\alpha$-spectral radius of the $k$-uniform hypergraphs, we completely prove that this conjecture is right for $0\leq\alpha<1$ and $m\geq 13$.

##### 6.Combinatorics of a Class of Completely Additive Arithmetic Functions

**Authors:**Hartosh Singh Bal

**Abstract:** The height $H(n)$ of $n$, is the smallest positive integer $i$ such that the $i$ th iterate of the totient function evaluated at $n$ is $1$. H. N. Shapiro determined that $H$ was almost completely additive. Building on the fact that the set of all numbers at a particular height reflect a partition structure the paper shows that all multi-partition structures correspond to a completely additive function. This includes the Matula number for rooted trees, as well as the sum of the primes function studied by Alladi and Erdos. We show the reordering of the natural numbers in each such partition structure is of importance and conjecture that the log of the asymptotic growth of the multi-partition is inversely related to the order of the growth of the associated additive function. We further provide computational evidence that the partition structure for large n tends to a lognormal distribution leading to insights into the distribution of primes at each height, which can thus be seen as a finite setting for probabilistic questions.

##### 7.Slow graph bootstrap percolation I: Cycles

**Authors:**David Fabian, Patrick Morris, Tibor Szabó

**Abstract:** Given a fixed graph $H$ and an $n$-vertex graph $G$, the $H$-bootstrap percolation process on $G$ is defined to be the sequence of graphs $G_i$, $i\geq 0$ which starts with $G_0:=G$ and in which $G_{i+1}$ is obtained from $G_i$ by adding every edge that completes a copy of $H$. We are interested in the maximum number of steps, over all $n$-vertex graphs $G$, that this process takes to stabilise. In the first of a series of papers exploring the behaviour of this function, denoted $M_H(n)$, and its dependence on certain properties of $H$, we investigate the case when $H$ is a cycle. We determine the running time precisely, giving the first infinite family of graphs $H$ for which an exact solution is known. The maximum running time of the $C_k$-bootstrap process is of the order $\log_{k-1}(n)$ for all $3\leq k\in \mathbb{N}$. Interestingly though, the function exhibits different behaviour depending on the parity of $k$ and the exact location of the values of $n$ for which $M_H(n)$ increases is determined by the Frobenius number of a certain numerical semigroup depending on $k$.

##### 8.On the Analysis of Boolean Functions and Fourier-Entropy-Influence Conjecture

**Authors:**Xiao Han

**Abstract:** This manuscript includes some classical results we select apart from the new results we've found on the Analysis of Boolean Functions and Fourier-Entropy-Influence conjecture. We try to ensure the self-completeness of this work so that readers could probably read it independently. Among the new results, what is the most remarkable is that we prove that the entropy of a boolean function $f$ could be bounded by $O(I(f))-O(\sum_{k\in [n]}I_k(f)\log I_k(f))$.

##### 9.Grading Structure for Derivations of Group Algebras

**Authors:**Andronick Arutyunov, Igor Zhiltsov

**Abstract:** In this paper we give a way of equipping the derivation algebra of a group algebra with the structure of a graded algebra. The derived group is used as the grading group. For the proof, the identification of the derivation with the characters of the adjoint action groupoid is used. These results also allow us to obtain the analogous structure of a graded algebra for outer derivations. A non-trivial graduation is obtained for all groups that are not perfect.

##### 10.New results on the 1-isolation number of graphs without short cycles

**Authors:**Yirui Huang, Gang Zhang, Xian'an Jin

**Abstract:** Let $G$ be a graph. A subset $D \subseteq V(G)$ is called a 1-isolating set of $G$ if $\Delta(G-N[D]) \leq 1$, that is, $G-N[D]$ consists of isolated edges and isolated vertices only. The $1$-isolation number of $G$, denoted by $\iota_1(G)$, is the cardinality of a smallest $1$-isolating set of $G$. In this paper, we prove that if $G \notin \{P_3,C_3,C_7,C_{11}\}$ is a connected graph of order $n$ without $6$-cycles, or without induced 5- and 6-cycles, then $\iota_1(G) \leq \frac{n}{4}$. Both bounds are sharp.

##### 11.On Rödl's Theorem for Cographs

**Authors:**Lior Gishboliner, Asaf Shapira

**Abstract:** A theorem of R\"odl states that for every fixed $F$ and $\varepsilon>0$ there is $\delta=\delta_F(\varepsilon)$ so that every induced $F$-free graph contains a vertex set of size $\delta n$ whose edge density is either at most $\varepsilon$ or at least $1-\varepsilon$. R\"odl's proof relied on the regularity lemma, hence it supplied only a tower-type bound for $\delta$. Fox and Sudakov conjectured that $\delta$ can be made polynomial in $\varepsilon$, and a recent result of Fox, Nguyen, Scott and Seymour shows that this conjecture holds when $F=P_4$. In fact, they show that the same conclusion holds even if $G$ contains few copies of $P_4$. In this note we give a short proof of a more general statement.

##### 12.Minimizing the number of edges in (C4, K1,k)-co-critical graphs

**Authors:**Gang Chen, Chenchen Ren, Zi-xia song

**Abstract:** Given graphs $H_1, H_2$, a \{red, blue\}-coloring of the edges of a graph $G$ is a critical coloring if $G$ has neither a red $H_1$ nor a blue $ H_2$. A non-complete graph $G$ is $(H_1, H_2)$-co-critical if $G$ admits a critical coloring, but $G+e$ has no critical coloring for every edge $e$ in the complement of $G$. Motivated by a conjecture of Hanson and Toft from 1987, we study the minimum number of edges over all $(C_4, K_{1,k})$-co-critical graphs on $n$ vertices. We show that for all $k \ge 2 $ and $ n \ge k +\lfloor \sqrt {k-1} \rfloor +2$, if $G$ is a $(C_4,K_{1,k})$-co-critical graph on $n$ vertices, then \[e(G) \ge \frac{(k+2)n}2-3- \frac{(k-1)(k+ \lfloor \sqrt {k-2}\rfloor)}2.\] Moreover, this linear bound is asymptotically best possible for all $k\ge3$ and $n\ge3k+4$. It is worth noting that our constructions for the case when $ k$ is even have at least three different critical colorings. For $k=2$, we obtain the sharp bound for the minimum number of edges of $(C_4, K_{1,2})$-co-critical graphs on $n\ge5 $ vertices by showing that all such graphs have at least $2n-3$ edges. Our proofs rely on the structural properties of $(C_4,K_{1,k})$-co-critical graphs and a result of Ollmann on the minimum number of edges of $C_4$-saturated graphs.

##### 1.Percolated stochastic block model via EM algorithm and belief propagation with non-backtracking spectra

**Authors:**Marianna Bolla, Daniel Zhou

**Abstract:** Whereas Laplacian and modularity based spectral clustering is apt to dense graphs, recent results show that for sparse ones, the non-backtracking spectrum is the best candidate to find assortative clusters of nodes. Here belief propagation in the sparse stochastic block model is derived with arbitrary given model parameters that results in a non-linear system of equations; with linear approximation, the spectrum of the non-backtracking matrix is able to specify the number $k$ of clusters. Then the model parameters themselves can be estimated by the EM algorithm. Bond percolation in the assortative model is considered in the following two senses: the within- and between-cluster edge probabilities decrease with the number of nodes and edges coming into existence in this way are retained with probability $\beta$. As a consequence, the optimal $k$ is the number of the structural real eigenvalues (greater than $\sqrt{c}$, where $c$ is the average degree) of the non-backtracking matrix of the graph. Assuming, these eigenvalues $\mu_1 >\dots > \mu_k$ are distinct, the multiple phase transitions obtained for $\beta$ are $\beta_i =\frac{c}{\mu_i^2}$; further, at $\beta_i$ the number of detectable clusters is $i$, for $i=1,\dots ,k$. Inflation-deflation techniques are also discussed to classify the nodes themselves, which can be the base of the sparse spectral clustering.

##### 2.On the Kohayakawa-Kreuter conjecture

**Authors:**Eden Kuperwasser, Wojciech Samotij, Yuval Wigderson

**Abstract:** Let us say that a graph $G$ is Ramsey for a tuple $(H_1,\dots,H_r)$ of graphs if every $r$-coloring of the edges of $G$ contains a monochromatic copy of $H_i$ in color $i$, for some $i \in [r]$. A famous conjecture of Kohayakawa and Kreuter, extending seminal work of R\"odl and Ruci\'nski, predicts the threshold at which the binomial random graph $G_{n,p}$ becomes Ramsey for $(H_1,\dots,H_r)$ asymptotically almost surely. In this paper, we resolve the Kohayakawa-Kreuter conjecture for almost all tuples of graphs. Moreover, we reduce its validity to the truth of a certain deterministic statement, which is a clear necessary condition for the conjecture to hold. All of our results actually hold in greater generality, when one replaces the graphs $H_1,\dots,H_r$ by finite families $\mathcal{H}_1,\dots,\mathcal{H}_r$. Additionally, we pose a natural (deterministic) graph-partitioning conjecture, which we believe to be of independent interest, and whose resolution would imply the Kohayakawa-Kreuter conjecture.

##### 3.Boolean dimension of a Boolean lattice

**Authors:**Marcin Briański, Jędrzej Hodor, Hoang La, Piotr Micek, Katzper Michno

**Abstract:** For every integer $n$ with $n \geq 6$, we prove that the Boolean dimension of a poset consisting of all the subsets of $\{1,\dots,n\}$ equipped with the inclusion relation is strictly less than $n$.

##### 4.A refinement of and a companion to MacMahon's partition identity

**Authors:**Matthew C. Russell

**Abstract:** We provide a refinement of MacMahon's partition identity on sequence-avoiding partitions, and use it to produce another mod 6 partition identity. In addition, we show that our technique also extends to cover Andrews's generalization of MacMahon's identity. Our proofs are bijective in nature, exploiting a theorem of Xiong and Keith.

##### 5.Proof of the Kohayakawa--Kreuter conjecture for the majority of cases

**Authors:**Candida Bowtell, Robert Hancock, Joseph Hyde

**Abstract:** For graphs $G, H_1,\dots,H_r$, write $G \to (H_1, \ldots, H_r)$ to denote the property that whenever we $r$-colour the edges of $G$, there is a monochromatic copy of $H_i$ in colour $i$ for some $i \in \{1,\dots,r\}$. Mousset, Nenadov and Samotij proved an upper bound on the threshold function for the property that $G_{n,p} \to (H_1,\dots,H_r)$, thereby resolving the $1$-statement of the Kohayakawa--Kreuter conjecture. We reduce the $0$-statement of the Kohayakawa--Kreuter conjecture to a natural deterministic colouring problem and resolve this problem for almost all cases, which in particular includes (but is not limited to) when $H_2$ is strictly $2$-balanced and either has density greater than $2$ or is not bipartite. In addition, we extend our reduction to hypergraphs, proving the colouring problem in almost all cases there as well.

##### 6.On bipartite coverings of graphs and multigraphs

**Authors:**Noga Alon

**Abstract:** A bipartite covering of a (multi)graph $G$ is a collection of bipartite graphs, so that each edge of $G$ belongs to at least one of them. The capacity of the covering is the sum of the numbers of vertices of these bipartite graphs. In this note we establish a (modest) strengthening of old results of Hansel and of Katona and Szemer\'edi, by showing that the capacity of any bipartite covering of a graph on $n$ vertices in which the maximum size of an independent set containing vertex number $i$ is $\alpha_i$, is at least $\sum_i \log_2 (n/\alpha_i).$ We also obtain slightly improved bounds for a recent result of Kim and Lee about the minimum possible capacity of a bipartite covering of complete multigraphs.

##### 7.Monotone links in DAHA and EHA

**Authors:**Pavel Galashin, Thomas Lam

**Abstract:** We define monotone links on a torus, obtained as projections of curves in the plane whose coordinates are monotone increasing. Using the work of Morton-Samuelson, to each monotone link we associate elements in the double affine Hecke algebra and the elliptic Hall algebra. In the case of torus knots (when the curve is a straight line), we recover symmetric function operators appearing in the rational shuffle conjecture. We show that the class of monotone links viewed as links in $\mathbb R^3$ coincides with the class of Coxeter links, studied by Oblomkov-Rozansky in the setting of the flag Hilbert scheme. When the curve satisfies a convexity condition, we recover positroid links that we previously studied. In the convex case, we conjecture that the associated symmetric functions are Schur positive, extending a recent conjecture of Blasiak-Haiman-Morse-Pun-Seelinger, and we speculate on the relation to Khovanov-Rozansky homology. Our constructions satisfy a skein recurrence where the base case consists of piecewise almost linear curves. We show that convex piecewise almost linear curves give rise to algebraic links.

##### 8.Antimagic Labelings of Forests

**Authors:**Johnny Sierra, Daphne Der-Fen Liu, Jessica Toy

**Abstract:** An antimagic labeling of a graph $G(V,E)$ is a bijection $f: E \to \{1,2, \dots, |E|\}$ so that $\sum_{e \in E(u)} f(e) \neq \sum_{e \in E(v)} f(e)$ holds for all $u, v \in V(G)$ with $u \neq v$, where $E(v)$ is the set of edges incident to $v$. We call $G$ antimagic if it admits an antimagic labeling. A forest is a graph without cycles; equivalently, every component of a forest is a tree. It was proved by Kaplan, Lev, and Roditty [2009], and by Liang, Wong, and Zhu [2014] that every tree with at most one vertex of degree-2 is antimagic. A major tool used in the proof is the zero-sum partition introduced by Kaplan, Lev, and Roditty [2009]. In this article, we provide an algorithmic representation for the zero-sum partition method and apply this method to show that every forest with at most one vertex of degree-2 is also antimagic.

##### 9.On higher multiplicity hyperplane and polynomial covers for symmetry preserving subsets of the hypercube

**Authors:**Arijit Ghosh, Chandrima Kayal, Soumi Nandi, S. Venkitesh

**Abstract:** Alon and F\"uredi (European J. Combin. 1993) gave a tight bound for the following hyperplane covering problem: find the minimum number of hyperplanes required to cover all points of the n-dimensional hypercube {0,1}^n except the origin. Their proof is among the early instances of the polynomial method, which considers a natural polynomial (a product of linear factors) associated to the hyperplane arrangement, and gives a lower bound on its degree, whilst being oblivious to the (product) structure of the polynomial. Thus, their proof gives a lower bound for a weaker polynomial covering problem, and it turns out that this bound is tight for the stronger hyperplane covering problem. In a similar vein, solutions to some other hyperplane covering problems were obtained, via solutions of corresponding weaker polynomial covering problems, in some special cases in the works of the fourth author (Electron. J. Combin. 2022), and the first three authors (Discrete Math. 2023). In this work, we build on these and solve a hyperplane covering problem for general symmetric sets of the hypercube, where we consider hyperplane covers with higher multiplicities. We see that even in this generality, it is enough to solve the corresponding polynomial covering problem. Further, this seems to be the limit of this approach as far as covering symmetry preserving subsets of the hypercube is concerned. We gather evidence for this by considering the class of blockwise symmetric sets of the hypercube (which is a strictly larger class than symmetric sets), and note that the same proof technique seems to only solve the polynomial covering problem.

##### 1.Inverting the General Order Sweep Map

**Authors:**Ying Wang, Guoce Xin, Yingrui Zhang

**Abstract:** Inspired by Thomas-Williams work on the modular sweep map, Garsia and Xin gave a simple algorithm for inverting the sweep map on rational $(m,n)$-Dyck paths for a coprime pairs $(m,n)$ of positive integers. We find their idea naturally extends for general Dyck paths. Indeed, we define a class of Order sweep maps on general Dyck paths, using different sweep orders on level $0$. We prove that each such Order sweep map is a bijection. This includes sweep map for general Dyck paths and incomplete general Dyck paths as special cases.

##### 2.Kruskal--Katona-Type Problems via Entropy Method

**Authors:**Ting-Wei Chao, Hung-Hsun Hans Yu

**Abstract:** In this paper, we investigate several extremal combinatorics problems that ask for the maximum number of copies of a fixed subgraph given the number of edges. We call this type of problems Kruskal--Katona-type problems. Most of the problems that will be discussed in this paper are related to the joints problem. There are two main results in this paper. First, we prove that, in a $3$-colored graph with $R$ red, $G$ green, $B$ blue edges, the number of rainbow triangles is at most $\sqrt{2RGB}$, which is sharp. Second, we give a generalization of the Kruskal--Katona theorem that implies many other previous generalizations. Both arguments use the entropy method, and the main innovation lies in a more clever argument that improves bounds given by Shearer's inequality.

##### 3.Tight Bound and Structural Theorem for Joints

**Authors:**Ting-Wei Chao, Hung-Hsun Hans Yu

**Abstract:** A joint of a set of lines $\mathcal{L}$ in $\mathbb{F}^d$ is a point that is contained in $d$ lines with linearly independent directions. The joints problem asks for the maximum number of joints that are formed by $L$ lines. Guth and Katz showed that the number of joints is at most $O(L^{3/2})$ in $\mathbb{R}^3$ using polynomial method. This upper bound is met by the construction given by taking the joints and the lines to be all the $d$-wise intersections and all the $(d-1)$-wise intersections of $M$ hyperplanes in general position. Furthermore, this construction is conjectured to be optimal. In this paper, we verify the conjecture and show that this is the only optimal construction by using a more sophisticated polynomial method argument. This is the first tight bound and structural theorem obtained using this method. We also give a new definition of multiplicity that strengthens the main result of a previous work by Tidor, Zhao and the second author. Lastly, we include some discussion on the constants for the joints of varieties problem.

##### 4.Catching a robber on a random $k$-uniform hypergraph

**Authors:**Joshua Erde, Mihyun Kang, Florian Lehner, Bojan Mohar, Dominik Schmid

**Abstract:** The game of \emph{Cops and Robber} is usually played on a graph, where a group of cops attempt to catch a robber moving along the edges of the graph. The \emph{cop number} of a graph is the minimum number of cops required to win the game. An important conjecture in this area, due to Meyniel, states that the cop number of an $n$-vertex connected graph is $O(\sqrt{n})$. In 2016, Pra{\l}at and Wormald [Meyniel's conjecture holds for random graphs, Random Structures Algorithms. 48 (2016), no. 2, 396-421. MR3449604] showed that this conjecture holds with high probability for random graphs above the connectedness threshold. Moreoever, {\L}uczak and Pra{\l}at [Chasing robbers on random graphs: Zigzag theorem, Random Structures Algorithms. 37 (2010), no. 4, 516-524. MR2760362] showed that on a $\log$-scale the cop number demonstrates a surprising \emph{zigzag} behaviour in dense regimes of the binomial random graph $G(n,p)$. In this paper, we consider the game of Cops and Robber on a hypergraph, where the players move along hyperedges instead of edges. We show that with high probability the cop number of the $k$-uniform binomial random hypergraph $G^k(n,p)$ is $O\left(\sqrt{\frac{n}{k}}\, \log n \right)$ for a broad range of parameters $p$ and $k$ and that on a $\log$-scale our upper bound on the cop number arises as the minimum of \emph{two} complementary zigzag curves, as opposed to the case of $G(n,p)$. Furthermore, we conjecture that the cop number of a connected $k$-uniform hypergraph on $n$ vertices is $O\left(\sqrt{\frac{n}{k}}\,\right)$.

##### 5.Three remarks on $\mathbf{W_2}$ graphs

**Authors:**Carl Feghali, Malory Marin

**Abstract:** Let $k \geq 1$. A graph $G$ is $\mathbf{W_k}$ if for any $k$ pairwise disjoint independent vertex subsets $A_1, \dots, A_k$ in $G$, there exist $k$ pairwise disjoint maximum independent sets $S_1, \dots, S_k$ in $G$ such that $A_i \subseteq S_i$ for $i \in [k]$. Recognizing $\mathbf{W_1}$ graphs is co-NP-hard, as shown by Chv\'atal and Hartnell (1993) and, independently, by Sankaranarayana and Stewart (1992). Extending this result and answering a recent question of Levit and Tankus, we show that recognizing $\mathbf{W_k}$ graphs is co-NP-hard for $k \geq 2$. On the positive side, we show that recognizing $\mathbf{W_k}$ graphs is, for each $k\geq 2$, FPT parameterized by clique-width and by tree-width. Finally, we construct graphs $G$ that are not $\mathbf{W_2}$ such that, for every vertex $v$ in $G$ and every maximal independent set $S$ in $G - N[v]$, the largest independent set in $N(v) \setminus S$ consists of a single vertex, thereby refuting a conjecture of Levit and Tankus.

##### 6.Larger matchings and independent sets in regular uniform hypergraphs of high girth

**Authors:**Deepak Bal, Patrick Bennett

**Abstract:** In this note we analyze two algorithms, one for producing a matching and one for an independent set, on $k$-uniform $d$-regular hypergraphs of large girth. As a result we obtain new lower bounds on the size of a maximum matching or independent set in such hypergraphs.

##### 7.Disproof of a conjecture on the minimum spectral radius and the domination number

**Authors:**Yarong Hu, Zhenzhen Lou, Qiongxiang Huang

**Abstract:** Let $G_{n,\gamma}$ be the set of all connected graphs on $n$ vertices with domination number $\gamma$. A graph is called a minimizer graph if it attains the minimum spectral radius among $G_{n,\gamma}$. Very recently, Liu, Li and Xie [Linear Algebra and its Applications 673 (2023) 233--258] proved that the minimizer graph over all graphs in $\mathbb{G}_{n,\gamma}$ must be a tree. Moreover, they determined the minimizer graph among $G_{n,\lfloor\frac{n}{2}\rfloor}$ for even $n$, and posed the conjecture on the minimizer graph among $G_{n,\lfloor\frac{n}{2}\rfloor}$ for odd $n$. In this paper, we disprove the conjecture and completely determine the unique minimizer graph among $G_{n,\lfloor\frac{n}{2}\rfloor}$ for odd $n$.

##### 8.Nonabelian partial difference sets constructed using abelian techniques

**Authors:**James Davis, John Polhill, Ken Smith, Eric Swartz

**Abstract:** A $(v,k,\lambda, \mu)$-partial difference set (PDS) is a subset $D$ of a group $G$ such that $|G| = v$, $|D| = k$, and every nonidentity element $x$ of $G$ can be written in either $\lambda$ or $\mu$ different ways as a product $gh^{-1}$, depending on whether or not $x$ is in $D$. Assuming the identity is not in $D$ and $D$ is inverse-closed, the corresponding Cayley graph ${\rm Cay}(G,D)$ will be strongly regular. Partial difference sets have been the subject of significant study, especially in abelian groups, but relatively little is known about PDSs in nonabelian groups. While many techniques useful for abelian groups fail to translate to a nonabelian setting, the purpose of this paper is to show that examples and constructions using abelian groups can be modified to generate several examples in nonabelian groups. In particular, in this paper we use such techniques to construct the first known examples of PDSs in nonabelian groups of order $q^{2m}$, where $q$ is a power of an odd prime $p$ and $m \ge 2$. The groups constructed can have exponent as small as $p$ or as large as $p^r$ in a group of order $p^{2r}$. Furthermore, we construct what we believe are the first known Paley-type PDSs in nonabelian groups and what we believe are the first examples of Paley-Hadamard difference sets in nonabelian groups, and, using analogues of product theorems for abelian groups, we obtain several examples of each. We conclude the paper with several possible future research directions.

##### 9.Minors of matroids represented by sparse random matrices over finite fields

**Authors:**Pu Gao, Peter Nelson

**Abstract:** Consider a random $n\times m$ matrix $A$ over the finite field of order $q$ where every column has precisely $k$ nonzero elements, and let $M[A]$ be the matroid represented by $A$. In the case that q=2, Cooper, Frieze and Pegden (RS\&A 2019) proved that given a fixed binary matroid $N$, if $k\ge k_N$ and $m/n\ge d_N$ where $k_N$ and $d_N$ are sufficiently large constants depending on N, then a.a.s. $M[A]$ contains $N$ as a minor. We improve their result by determining the sharp threshold (of $m/n$) for the appearance of a fixed matroid $N$ as a minor of $M[A]$, for every $k\ge 3$, and every finite field.

##### 1.Spectral Turán-type problems on sparse spanning graphs

**Authors:**Lele Liu, Bo Ning

**Abstract:** Let $F$ be a graph and $\SPEX (n, F)$ be the class of $n$-vertex graphs which attain the maximum spectral radius and contain no $F$ as a subgraph. Let $\EX (n, F)$ be the family of $n$-vertex graphs which contain maximum number of edges and no $F$ as a subgraph. It is a fundamental problem in spectral extremal graph theory to characterize all graphs $F$ such that $\SPEX (n, F)\subseteq \EX (n, F)$ when $n$ is sufficiently large. Establishing the conjecture of Cioab\u{a}, Desai and Tait [European J. Combin., 2022], Wang, Kang, and Xue [J. Combin. Theory Ser. B, 2023] prove that: for any graph $F$ such that the graphs in $\EX (n, F)$ are Tur\'{a}n graphs plus $O(1)$ edges, $\SPEX (n, F)\subseteq \EX (n, F)$ for sufficiently large $n$. In this paper, we prove that $\SPEX (n, F)\subseteq \EX (n, F)$ for sufficiently large $n$, where $F$ is an $n$-vertex graph with no isolated vertices and $\Delta (F) \leq \sqrt{n}/40$. We also prove a signless Laplacian spectral radius version of the above theorem. These results give new contribution to the open problem mentioned above, and can be seen as spectral analogs of a theorem of Alon and Yuster [J. Combin. Theory Ser. B, 2013]. Furthermore, as immediate corollaries, we have tight spectral conditions for the existence of several classes of special graphs, including clique-factors, $k$-th power of Hamilton cycles and $k$-factors in graphs. The first special class of graphs gives a positive answer to a problem of Feng, and the second one extends a previous result of Yan et al.

##### 2.Variations on the Bollobás set-pair theorem

**Authors:**Gábor Hegedüs, Péter Frankl

**Abstract:** Let $X$ be an $n$-element set. A set-pair system $\mbox{$\cal P$}=\{(A_i,B_i)\}_{1\leq i\leq m}$ is a collection of pairs of disjoint subsets of $X$. It is called skew Bollob\'as system if $A_i\cap B_j\neq \emptyset$ for all $1\leq i<j \leq m$. The best possible inequality $$ \sum_{i=1}^m \frac{1}{{|A_i|+|B_i| \choose |A_i|}}\leq n+1. $$ is established along with some more results of similar flavor.

##### 3.Construction of graphs being determined by their generalized Q-spectra

**Authors:**Gui-Xian Tian, Jun-Xing Wu, Shu-Yu Cui, Hui-Lu Sun

**Abstract:** Given a graph $G$ on $n$ vertices, its adjacency matrix and degree diagonal matrix are represented by $A(G)$ and $D(G)$, respectively. The $Q$-spectrum of $G$ consists of all the eigenvalues of its signless Laplacian matrix $Q(G)=A(G)+D(G)$ (including the multiplicities). A graph $G$ is known as being determined by its generalized $Q$-spectrum ($DGQS$ for short) if, for any graph $H$, $H$ and $G$ have the same $Q$-spectrum and so do their complements, then $H$ is isomorphic to $G$. In this paper, we present a method to construct $DGQS$ graphs. More specifically, let the matrix $W_{Q}(G)=\left [e,Qe,\dots ,Q^{n-1}e \right ]$ ($e$ denotes the all-one column vector ) be the $Q$-walk matrix of $G$. It is shown that $G\circ P_{k}$ ($k=2,3$) is $DGQS$ if and only if $G$ is $DGQS$ for some specific graphs. This also provides a way to construct $DGQS$ graphs with more vertices by using $DGQS$ graphs with fewer vertices. At the same time, we also prove that $G\circ P_{2}$ is still $DGQS$ under specific circumstances. In particular, on the basis of the above results, we obtain an infinite sequences of $DGQS$ graphs $G\circ P_{k}^{t}$ ($k=2,3;t\ge 1$) for some specific $DGQS$ graph $G$.

##### 4.The last patch for classifying shuffle groups

**Authors:**Junyang Zhang

**Abstract:** Divide a deck of $kn$ cards into $k$ equal piles and place them from left to right. The standard shuffle $\sigma$ is performed by picking up the top cards one by one from left to right and repeating until all cards have been picked up. For every permutation $\tau$ of the $k$ piles, use $\rho_{\tau}$ to denote the induced permutation on the $kn$ cards. The shuffle group $G_{k,kn}$ is generated by $\sigma$ and the $k!$ permutations $\rho_{\tau}$. It was conjectured by Cohen et al in 2005 that the shuffle group $G_{k,kn}$ contains $A_{kn}$ if $k\geq3$, $(k,n)\ne\{4,2^f\}$ for any positive integer $f$ and $n$ is not a power of $k$. Very recently, Xia, Zhang and Zhu reduced the proof of the conjecture to that of the $2$-transitivity of the shuffle group and then proved the conjecture under the condition that $k\ge4$ or $k\nmid n$. In this paper, we proved that the group $G_{3,3n}$ is $2$-transitive for any positive integer $n$ which is a multiple of $3$ but not a power of $3$. This result leads to the complete classification of the shuffle groups $G_{k,kn}$ for all $k\ge2$ and $n\ge1$.

##### 5.Induced subgraph density. V. All paths approach Erdos-Hajnal

**Authors:**Tung Nguyen, Alex Scott, Paul Seymour

**Abstract:** The Erd\H{o}s-Hajnal conjecture says that for every graph $H$, there exists $c>0$ such that every $H$-free graph $G$ has a clique or stable set of size at least $2^{c\log|G|}$ (a graph is ``$H$-free'' if no induced subgraph is isomorphic to $H$). The conjecture is known when $H$ is a path with at most four vertices, but remains open for longer paths. For the five-vertex path, Blanco and Buci\'c recently proved a bound of $2^{c(\log |G|)^{2/3}}$; for longer paths, the best existing bound is $2^{c(\log|G|\log\log|G|)^{1/2}}$. We prove a much stronger result: for any path $P$, every $P$-free graph $G$ has a clique or stable set of size at least $2^{(\log |G|)^{1-o(1)}}$. We strengthen this further, weakening the hypothesis that $G$ is $P$-free by a hypothesis that $G$ does not contain ``many'' copies of $P$, and strengthening the conclusion, replacing the large clique or stable set outcome with a ``near-polynomial'' version of Nikiforov's theorem.

##### 6.Correspondence coloring of random graphs

**Authors:**Zdenek Dvorak, Liana Yepremyan

**Abstract:** We show that Erd\H{o}s-R\'enyi random graphs $G(n,p)$ with constant density $p<1$ have correspondence chromatic number $O(n/\sqrt{\log n})$; this matches a prediction from linear Hadwiger's conjecture for correspondence coloring. The proof follows from a simple sufficient condition for correspondence colorability in terms of the numbers of independent sets.

##### 1.Recognition of chordal graphs and cographs which are Cover-Incomparability graphs

**Authors:**Arun Anil, Manoj Changat

**Abstract:** Cover-Incomparability graphs (C-I graphs) are an interesting class of graphs from posets. A C-I graph is a graph from a poset $P=(V,\le)$ with vertex set $V$, and the edge-set is the union of edge sets of the cover graph and the incomparability graph of the poset. The recognition of the C-I graphs is known to be NP-complete (Maxov\'{a} et al., Order 26(3), 229--236(2009)). In this paper, we prove that chordal graphs having at most two independent simplicial vertices are exactly the chordal graphs which are also C-I graphs. A similar result is obtained for cographs as well. Using the structural results of these graphs, we derive linear time recognition algorithms for chordal graphs and cographs, which are C-I graphs.

##### 2.HBS Tilings extended: State of the art and novel observations

**Authors:**Carole Porrier

**Abstract:** Penrose tilings are the most famous aperiodic tilings, and they have been studied extensively. In particular, patterns composed with hexagons ($H$), boats ($B$) and stars ($S$) were soon exhibited and many physicists published on what they later called $HBS$ tilings, but no article or book combines all we know about them. This work is done here, before introducing new decorations and properties including explicit substitutions. For the latter, the star comes in three versions so we have 5 prototiles in what we call the Star tileset. Yet this set yields exactly the strict $HBS$ tilings formed using 3 tiles decorated with either the usual decorations (arrows) or Ammann bar markings for instance. Another new tileset called Gemstones is also presented, derived from the Star tileset.

##### 3.$3$-Neighbor bootstrap percolation on grids

**Authors:**Jaka Hedžet, Michael A. Henning

**Abstract:** Given a graph $G$ and assuming that some vertices of $G$ are infected, the $r$-neighbor bootstrap percolation rule makes an uninfected vertex $v$ infected if $v$ has at least $r$ infected neighbors. The $r$-percolation number, $m(G, r)$, of $G$ is the minimum cardinality of a set of initially infected vertices in $G$ such that after continuously performing the $r$-neighbor bootstrap percolation rule each vertex of $G$ eventually becomes infected. In this paper, we consider the $3$-bootstrap percolation number of grids with fixed widths. If $G$ is the cartesian product $P_3 \square P_m$ of two paths of orders~$3$ and $m$, we prove that $m(G,3)=\frac{3}{2}(m+1)-1$, when $m$ is odd, and $m(G,3)=\frac{3}{2}m +1$, when $m$ is even. Moreover if $G$ is the cartesian product $P_5 \square P_m$, we prove that $m(G,3)=2m+2$, when $m$ is odd, and $m(G,3)=2m+3$, when $m$ is even. If $G$ is the cartesian product $P_4 \square P_m$, we prove that $m(G,3)$ takes on one of two possible values, namely $m(G,3) = \lfloor \frac{5(m+1)}{3} \rfloor + 1$ or $m(G,3) = \lfloor \frac{5(m+1)}{3} \rfloor + 2$.

##### 4.Alder-type partition inequality of levels 2 and 3

**Authors:**Haein Cho, Soon-Yi Kang, Byungchan Kim

**Abstract:** A Known Alder-type partition inequality of level $a$, which involves the second Rogers-Ramanujan identity when the level $a$ is 2, states that the number of partitions of $n$ into parts differing by at least $d$ with the smallest part being at least $a$ is greater than or equal to that of partitions of $n$ into parts congruent to $\pm a \pmod{d+3}$, excluding the part $d+3-a$. In this paper, we prove levels 2 and 3 Alder-type partition inequalities for all but a finite number of $d$, without requiring the exclusion of the part $d+3-a$ in the latter partition.

##### 5.Metallic cubes

**Authors:**Tomislav Došlić, Luka Podrug

**Abstract:** We study a recursively defined two-parameter family of graphs which generalize Fibonacci cubes and Pell graphs and determine their basic structural and enumerative properties. In particular, we show that all of them are induced subgraphs of hypercubes and present their canonical decompositions. Further, we compute their metric invariants and establish some Hamiltonicity properties. We show that the new family inherits many useful properties of Fibonacci cubes and hence could be interesting for potential applications. We also compute the degree distribution, opening thus the way for computing many degree-based topological invariants. Several possible directions of further research are discussed in the concluding section.

##### 6.Cliqueful graphs as a means of calculating the maximal number of maximum cliques of simple graphs

**Authors:**Dániel Pfeifer

**Abstract:** A simple graph on $n$ vertices may contain a lot of maximum cliques. But how many can it potentially contain? We will show that the maximum number of maximum cliques is taken over so-called cliqueful graphs, more specifically, later we will show that it is taken over saturated composite cliqueful graphs, if $n \ge 15$. Using this we will show that the graph that contains $3^{\lfloor n/3 \rfloor}c$ maxcliques has the most number of maxcliques on $n$ vertices, where $c\in\{1,\frac{4}{3},2\}$, depending on $n \text{ mod } 3$.

##### 7.Higher-dimensional cubical sliding puzzles

**Authors:**Moritz Beyer, Stefano Mereta, Érika Roldán, Peter Voran

**Abstract:** We introduce higher-dimensional cubical sliding puzzles that are inspired by the classical 15 Puzzle from the 1880s. In our puzzles, on a $d$-dimensional cube, a labeled token can be slid from one vertex to another if it is topologically free to move on lower-dimensional faces. We analyze the solvability of these puzzles by studying how the puzzle graph changes with the number of labeled tokens vs empty vertices. We give characterizations of the different regimes ranging from being completely stuck (and thus all puzzles unsolvable) to having only one giant component where almost all puzzles can be solved. For the Cube, the Tesseract, and the Penteract ($5$-dimensional cube) we have implemented an algorithm to completely analyze their solvability and we provide specific puzzles for which we know the minimum number of moves needed to solve them.

##### 8.On the distribution of the entries of a fixed-rank random matrix over a finite field

**Authors:**Carlo Sanna

**Abstract:** Let $r > 0$ be an integer, let $\mathbb{F}_q$ be a finite field of $q$ elements, and let $\mathcal{A}$ be a nonempty proper subset of $\mathbb{F}_q$. Moreover, let $\mathbf{M}$ be a random $m \times n$ rank-$r$ matrix over $\mathbb{F}_q$ taken with uniform distribution. We prove, in a precise sense, that, as $m, n \to +\infty$ and $r,q,\mathcal{A}$ are fixed, the number of entries of $\mathbf{M}$ that belong to $\mathcal{A}$ approaches a normal distribution.

##### 9.On colorings of hypergraphs embeddable in $\mathbb{R}^d$

**Authors:**Seunghun Lee, Eran Nevo

**Abstract:** The \textit{(weak) chromatic number} of a hypergraph $H$, denoted by $\chi(H)$, is the smallest number of colors required to color the vertices of $H$ so that no hyperedge of $H$ is monochromatic. For every $2\le k\le d+1$, denote by $\chi_L(k,d)$ (resp. $\chi_{PL}(k,d)$) the supremum $\sup_H \chi(H)$ where $H$ runs over all finite $k$-uniform hypergraphs such that $H$ forms the collection of maximal faces of a simplicial complex that is linearly (resp. PL) embeddable in $\mathbb{R}^d$. Following the program by Heise, Panagiotou, Pikhurko and Taraz, we improve their results as follows: For $d \geq 3$, we show that A. $\chi_L(k,d)=\infty$ for all $2\le k\le d$, B. $\chi_{PL}(d+1,d)=\infty$ and C. $\chi_L(d+1,d)\ge 3$ for all odd $d\ge 3$. As an application, we extend the results by Lutz and M\o ller on the weak chromatic number of the $s$-dimensional faces in the triangulations of a fixed triangulable $d$-manifold $M$: D. $\chi_s(M)=\infty$ for $1\leq s \leq d$.

##### 10.Generating functions of non-backtracking walks on weighted digraphs: radius of convergence and Ihara's theorem

**Authors:**Vanni Noferini, María C. Quintana

**Abstract:** It is known that the generating function associated with the enumeration of non-backtracking walks on finite graphs is a rational matrix-valued function of the parameter; such function is also closely related to graph-theoretical results such as Ihara's theorem and the zeta function on graphs. In [P. Grindrod, D. J. Higham, V. Noferini, The deformed graph Laplacian and its application to network centrality analysis, SIAM J. Matrix Anal. Appl. 39(1), 310--341, 2018], the radius of convergence of the generating function was studied for simple (i.e., undirected, unweighted and with no loops) graphs, and shown to depend on the number of cycles in the graph. In this paper, we use technologies from the theory of polynomial and rational matrices to greatly extend these results by studying the radius of convergence of the corresponding generating function for general, possibly directed and/or weighted, graphs. We give an analogous characterization of the radius of convergence for directed unweighted graphs, showing that it depends on the number of cycles in the undirectization of the graph. For weighted graphs, we provide for the first time an exact formula for the radius of convergence, improving a previous result that exhibited a lower bound. Finally, we consider also backtracking-downweighted walks on unweighted digraphs, and we prove a version of Ihara's theorem in that case.

##### 11.Removing induced powers of cycles from a graph via fewest edits

**Authors:**Amarja Kathapurkar, Richard Mycroft

**Abstract:** What is the minimum proportion of edges which must be added to or removed from a graph of density $p$ to eliminate all induced cycles of length $h$? The maximum of this quantity over all graphs of density $p$ is measured by the edit distance function, $\text{ed}_{\text{Forb}(C_h)}(p)$, a function which provides a natural metric between graphs and hereditary properties. Martin determined $\text{ed}_{\text{Forb}(C_h)}(p)$ for all $p \in [0,1]$ when $h \in \{3, \ldots, 9\}$ and determined $\text{ed}_{\text{Forb}(C_{10})}(p)$ for $p \in [1/7, 1]$. Peck determined $\text{ed}_{\text{Forb}(C_h)}(p)$ for all $p \in [0,1]$ for odd cycles, and for $p \in [ 1/\lceil h/3 \rceil, 1]$ for even cycles. In this paper, we fully determine the edit distance function for $C_{10}$ and $C_{12}$. Furthermore, we improve on the result of Peck for even cycles, by determining $\text{ed}_{\text{Forb}(C_h)}(p)$ for all $p \in [p_0, 1/\lceil h/3 \rceil ]$, where $p_0 \leq c/h^2$ for a constant $c$. More generally, if $C_h^t$ is the $t$-th power of the cycle $C_h$, we determine $\text{ed}_{\text{Forb}(C_h^t)}(p)$ for all $p \geq p_0$ in the case when $(t+1) \mid h$, thus improving on earlier work of Berikkyzy, Martin and Peck.

##### 12.Representing matroids via pasture morphisms

**Authors:**Tianyi Zhang, Justin Chen

**Abstract:** Using the framework of pastures and foundations of matroids developed by Baker-Lorscheid, we give algorithms to: (i) compute the foundation of a matroid, and (ii) compute all morphisms between two pastures. Together, these provide an efficient method of solving many questions of interest in matroid representations, including orientability, non-representability, and computing all representations of a matroid over a finite field.

##### 1.Covering triangular grids with multiplicity

**Authors:**Abdul Basit, Alexander Clifton, Paul Horn

**Abstract:** Motivated by classical work of Alon and F\"uredi, we introduce and address the following problem: determine the minimum number of affine hyperplanes in $\mathbb{R}^d$ needed to cover every point of the triangular grid $T_d(n) := \{(x_1,\dots,x_d)\in\mathbb{Z}_{\ge 0}^d\mid x_1+\dots+x_d\le n-1\}$ at least $k$ times. For $d = 2$, we solve the problem exactly for $k \leq 4$, and obtain a partial solution for $k > 4$. We also obtain an asymptotic formula (in $n$) for all $d \geq k - 2$. The proofs rely on combinatorial arguments and linear programming.

##### 2.Adjacency spectra of some subdivision hypergraphs

**Authors:**Anirban Banerjee, Arpita Das

**Abstract:** Here, we define a subdivision operation for a hypergraph and compute all the eigenvalues of the subdivision of regular and certain non-regular hypergraphs. In non-regular hypergraphs, we investigate the power of regular graphs, various types of hyperflowers, and the squid-like hypergraph. Using our subdivision operation, we also show how to construct non-regular non-isomorphic cospectral hypergraphs.

##### 3.k-Pell graphs

**Authors:**Vesna Iršič, Sandi Klavžar, Elif Tan

**Abstract:** In this paper, $k$-Pell graphs $\Pi _{n,k}$, $k\ge 2$, are introduced such that $\Pi _{n,2}$ are the Pell graphs defined earlier by Munarini. Several metric, enumerative, and structural properties of these graphs are established. The generating function of the number of edges of $\Pi _{n,k}$ and the generating function of its cube polynomial are determined. The center of $\Pi _{n,k}$ is explicitly described; if $k$ is even, then it induces the Fibonacci cube $\Gamma_{n}$. It is also shown that $\Pi _{n,k}$ is a median graph, and that $\Pi _{n,k}$ embeds into some Fibonacci cube.

##### 4.Pattern avoidance and smoothness of Hessenberg Schubert varieties

**Authors:**Soojin Cho, JiSun Huh, Seonjeong Park

**Abstract:** A \emph{Hessenberg Schubert variety} is the closure of a Schubert cell inside a given Hessenberg variety. We consider the smoothness of Hessenberg Schubert varieties of regular semisimple Hessenberg varieties of type $A$ in this paper. We use known combinatorial characterizations of torus fixed points of a Hessenberg Schubert variety $\Omega_{w, h}$ to find a necessary condition for $\Omega_{w, h}$ to be smooth, in terms of the pattern avoidance of the permutation $w$. First, we show useful theorems regarding the structure of the subposet of the Bruhat order induced by the torus fixed points of $\Omega_{w, h}$. Then we apply them to prove that the regularity of the associated graph, which is known to be a necessary condition for the smoothness of $\Omega_{w, h}$, is completely characterized by the avoidance of the patterns we found.

##### 5.Quasi-coincidence of cluster structures on positroid varieties

**Authors:**Matthew Pressland

**Abstract:** By work of a number of authors, beginning with Scott and culminating with Galashin and Lam, the coordinate rings of positroid varieties in the Grassmannian carry cluster algebra structures. In fact, they typically carry many such structures, the two best understood being the source-labelled and target-labelled structures, referring to how the initial cluster is computed from a Postnikov diagram or plabic graph. In this article we show that these two cluster algebra structures quasi-coincide, meaning in particular that a cluster variable in one structure may be expressed in the other structure as the product of a cluster variable and a Laurent monomial in the frozen variables. This resolves a conjecture attributed to Muller and Speyer from 2017. The proof depends critically on categorification: of the relevant cluster algebra structures by the author, of perfect matchings and twists by the author with \c{C}anak\c{c}{\i} and King, and of quasi-equivalences of cluster algebras by Fraser-Keller. By similar techniques, we also show that Muller-Speyer's left twist map is a quasi-cluster equivalence from the target-labelled structure to the source-labelled structure.

##### 6.Metric Location in Pseudotrees: A survey and new results

**Authors:**José Cáceres, Ignacio M. Pelayo

**Abstract:** The aim of this paper is to revise the literature on different metric locations in the families of paths, cycles, trees and unicyclic graphs, as well as, provide several new results on that matter.

##### 7.Eigenvalue Bounds for Sum-Rank-Metric Codes

**Authors:**Aida Abiad, Antonina P. Kharmova, Alberto Ravagnani

**Abstract:** We consider the problem of deriving upper bounds on the parameters of sum-rank-metric codes, with focus on their dimension and block length. The sum-rank metric is a combination of the Hamming and the rank metric, and most of the available techniques to investigate it seem to be unable to fully capture its hybrid nature. In this paper, we introduce a new approach based on sum-rank-metric graphs, in which the vertices are tuples of matrices over a finite field, and where two such tuples are connected when their sum-rank distance is equal to one. We establish various structural properties of sum-rank-metric graphs and combine them with eigenvalue techniques to obtain bounds on the cardinality of sum-rank-metric codes. The bounds we derive improve on the best known bounds for several choices of the parameters. While our bounds are explicit only for small values of the minimum distance, they clearly indicate that spectral theory is able to capture the nature of the sum-rank-metric better than the currently available methods. They also allow us to establish new non-existence results for (possibly nonlinear) MSRD codes.

##### 8.On optimal constant weight codes derived from $ω$-circulant balanced generalized weighing matrices

**Authors:**Hadi Kharaghani, Thomas Pender, Vladimir D. Tonchev

**Abstract:** A family of $\omega$-circulant balanced weighing matrices with classical parameters is used for the construction of optimal constant weight codes over an alphabet of size $g+1$ and length $n=(q^m -1)/(q-1)$, where $q$ is an odd prime power, $m>1$, and $g$ is a divisor of $q-1$.

##### 9.Induced subgraphs and tree decompositions X. Towards logarithmic treewidth for even-hole-free graphs

**Authors:**Tara Abrishami, Bogdan Alecu, Maria Chudnovsky, Sepehr Hajebi, Sophie Spirkl

**Abstract:** A generalized $t$-pyramid is a graph obtained from a certain kind of tree (a subdivided star or a subdivided cubic caterpillar) and the line graph of a subdivided cubic caterpillar by identifying simplicial vertices. We prove that for every integer $t$ there exists a constant $c(t)$ such that every $n$-vertex even-hole-free graph with no clique of size $t$ and no induced subgraph isomorphic to a generalized $t$-pyramid has treewidth at most $c(t)\log{n}$. This settles a special case of a conjecture of Sintiari and Trotignon; this bound is also best possible for the class. It follows that several NP-hard problems such as Stable Set, Vertex Cover, Dominating Set and Coloring admit polynomial-time algorithms on this class of graphs.

##### 1.Note on cartesian product of some combinatorially rich sets

**Authors:**Pintu Debnath

**Abstract:** D. De, N. Hindman, and D. Strauss have introduced $C$-set in \cite{key-5}, satisfying the strong central set theorem. Using the algebraic structure of the Stone-\v{C}ech compactification of a discrete semigroup, N. Hindman and D. Strauss proved that the product of two $C$-sets is a $C$-set. In \cite{key-7}, S. Goswami has proved the same result using the elementary characterization of $C$-set. In this paper we prove that the product of two $C$-sets is a $C$-set, using the dynamical characterization of $C$-set.

##### 2.Weakly distance-regular circulants, I

**Authors:**Akihiro Munemasa, Kaishun Wang, Yuefeng Yang, Wenying Zhu

**Abstract:** We classify certain non-symmetric commutative association schemes. As an application, we determine all the weakly distance-regular circulants of one type of arcs by using Schur rings. We also give the classification of primitive weakly distance-regular circulants.

##### 3.Laplacian spectrum of weakly zero-divisor graph of the ring $\mathbb{Z}_{n}$

**Authors:**Mohd Shariq, Praveen Mathil, Jitender Kumar

**Abstract:** Let $R$ be a commutative ring with unity. The weakly zero-divisor graph $W\Gamma(R)$ of the ring $R$ is the simple undirected graph whose vertices are nonzero zero-divisors of $R$ and two vertices $x$, $y$ are adjacent if and only if there exists $r\in {\rm ann}(x)$ and $s \in {\rm ann}(y)$ such that $rs =0$. The zero-divisor graph of a ring is a spanning subgraph of the weakly zero-divisor graph. It is known that the zero-divisor graph of the ring $\mathbb{Z}_{{p^t}}$, where $p$ is a prime, is the Laplacian integral. In this paper, we obtain the Laplacian spectrum of the weakly zero-divisor graph $W\Gamma(\mathbb{Z}_{n})$ of the ring $\mathbb{Z}_{n}$ and show that $W\Gamma(\mathbb{Z}_{n})$ is Laplacian integral for arbitrary $n$.

##### 4.Scattered trinomials of $\mathbb{F}_{q^6}[X]$ in even characteristic

**Authors:**Daniele Bartoli, Giovanni Longobardi, Giuseppe Marino, Marco Timpanella

**Abstract:** In recent years, several families of scattered polynomials have been investigated in the literature. However, most of them only exist in odd characteristic. In [B. Csajb\'ok, G. Marino and F. Zullo: New maximum scattered linear sets of the projective line, Finite Fields Appl. 54 (2018), 133-150; G. Marino, M. Montanucci and F. Zullo: MRD-codes arising from the trinomial $x^q+x^{q^3}+cx^{q^5}\in\mathbb{F}_{q^6}[x]$, Linear Algebra Appl. 591 (2020), 99-114], the authors proved that the trinomial $f_c(X)=X^{q}+X^{q^{3}}+cX^{q^{5}}$ of $\mathbb{F}_{q^6}[X]$ is scattered under the assumptions that $q$ is odd and $c^2+c=1$. They also explicitly observed that this is false when $q$ is even. In this paper, we provide a different set of conditions on $c$ for which this trinomial is scattered in the case of even $q$. Using tools of algebraic geometry in positive characteristic, we show that when $q$ is even and sufficiently large, there are roughly $q^3$ elements $c \in \mathbb{F}_{q^6}$ such that $f_{c}(X)$ is scattered. Also, we prove that the corresponding MRD-codes and $\mathbb{F}_q$-linear sets of $\mathrm{PG}(1,q^6)$ are not equivalent to the previously known ones.

##### 5.Taylor is prime

**Authors:**Bertalan Bodor, Gergő Gyenizse, Miklós Maróti, László Zádori

**Abstract:** We study the Taylor varieties and obtain new characterizations of them via compatible reflexive digraphs. Based on our findings, we prove that in the lattice of interpretability types of varieties, the filter of the types of all Taylor varieties is prime.

##### 6.A note on the equidistribution of $3$-colour partitions

**Authors:**Joshua Males

**Abstract:** In this short note, we prove equidistribution results regarding three families of three-colour partitions recently introduced by Schlosser and Zhou. To do so, we prove an asymptotic for the infinite product $F_{a,c}(\zeta ; q) \coloneqq \prod_{n \geq 0} \left(1- \zeta q^{a+cn}\right)$ ($a,c \in \N$ with $0<a\leq c$ and $\zeta$ a root of unity) in certain cones in the right half-plane, which may be useful in studying similar problems. As a corollary, we obtain the asymptotic behaviour of the three-colour partition families at hand.

##### 1.Tilings of the Sphere by Congruent Pentagons IV: Edge Combination $a^4b$

**Authors:**Hoi Ping Luk, Min Yan

**Abstract:** We classify edge-to-edge tilings of the sphere by congruent almost equilateral pentagons, in which four edges have the same length. Together with our earlier classifications of edge-to-edge tilings of the sphere by congruent equilateral pentagons of other types, and our classification of edge-to-edge tilings of the sphere by congruent quadrilaterals or triangles, we complete the classification of edge-to-edge tilings of the sphere by congruent polygons.

##### 2.Graphs with isolation number equal to one third of the order

**Authors:**Magdalena Lemanska, Mercè Mora, María José Souto-Salorio

**Abstract:** A set $D$ of vertices of a graph $G$ is isolating if the set of vertices not in $D$ or with no neighbor in $D$ is independent. The isolation number of $G$, denoted by $\iota (G)$, is the minimum cardinality of an isolating set of $G$. It is known that $\iota (G)\le n/3$, if $G$ is a connected graph of order $n$, $n\ge 3$, distinct from $C_5$. The main result of this work is the characterisation of unicyclic and block graphs of order $n$ with isolating number equal to $n/3$. Moreover, we provide a family of general graphs attaining this upper bound on the isolation number.

##### 3.Full asymptotic expansion for orbit-summable quadrant walks and discrete polyharmonic functions

**Authors:**Andreas Nessmann

**Abstract:** Enumeration of walks with small steps in the quadrant has been a topic of great interest in combinatorics over the last few years. In this article, it is shown how to compute exact asymptotics of the number of such walks with fixed start- and endpoints for orbit-summable models with finite group, up to arbitrary precision. The resulting representation greatly resembles one conjectured by Chapon, Fusy and Raschel for walks starting from the origin (AofA 2020), differing only in terms appearing due to the periodicity of the model. We will see that the dependency on start- and endpoint is given by discrete polyharmonic functions, which are solutions of $\triangle^n v=0$ for a discretisation $\triangle$ of a Laplace-Beltrami operator. They can be decomposed into a sum of products of lower order polyharmonic functions of either the start- or the endpoint only, which leads to a partial extension of a recent theorem by Denisov and Wachtel (Ann. Prob. 43.3).

##### 4.Competition graphs of degree bounded digraphs

**Authors:**Hojin Chu, Suh-Ryung Kim

**Abstract:** If each vertex of an acyclic digraph has indegree at most $i$ and outdegree at most $j$, then it is called an $(i,j)$ digraph, which was introduced by Hefner~{\it et al.}~(1991). Whereas Hefner~{\it et al.} characterized $(i,j)$ digraphs whose competition graphs are interval, characterizing the competition graphs of $(i,j)$ digraphs is not an easy task. In this paper, we introduce the concept of $\langle i,j \rangle$ digraphs, which relax the acyclicity condition of $(i,j)$ digraphs, and study their competition graphs. By doing so, we obtain quite meaningful results. Firstly, we give a necessary and sufficient condition for a loopless graph being an $\langle i,j \rangle$ competition graph for some positive integers $i$ and $j$. Then we study on an $\langle i,j \rangle$ competition graph being chordal and present a forbidden subdigraph characterization. Finally, we study the family of $\langle i,j \rangle$ competition graphs, denoted by $\mathcal{G}_{\langle i,j \rangle}$, and identify the set containment relation on $\{\mathcal{G}_{\langle i,j \rangle}\colon\, i,j \ge 1\}$.

##### 5.On commutative association schemes and associated (directed) graphs

**Authors:**Giusy Monzillo, Safet Penjić

**Abstract:** Let ${\cal M}$ denote the Bose--Mesner algebra of a commutative $d$-class association scheme ${\mathfrak X}$ (not necessarily symmetric), and $\Gamma$ denote a (strongly) connected (directed) graph with adjacency matrix $A$. Under the assumption that $A$ belongs to ${\cal M}$, in this paper, we describe the combinatorial structure of $\Gamma$. Among else, we show that, if ${\mathfrak X}$ is a commutative $3$-class association scheme that is not an amorphic symmetric scheme, then we can always find a (directed) graph $\Gamma$ such that the adjacency matrix $A$ of $\Gamma$ generates the Bose--Mesner algebra ${\cal M}$ of ${\mathfrak X}$.

##### 1.Mutual-visibility in distance-hereditary graphs: a linear-time algorithm

**Authors:**Serafino Cicerone, Gabriele Di Stefano

**Abstract:** The concept of mutual-visibility in graphs has been recently introduced. If $X$ is a subset of vertices of a graph $G$, then vertices $u$ and $v$ are $X$-visible if there exists a shortest $u,v$-path $P$ such that $V(P)\cap X \subseteq \{u, v\}$. If every two vertices from $X$ are $X$-visible, then $X$ is a mutual-visibility set. The mutual-visibility number of $G$ is the cardinality of a largest mutual-visibility set of $G$. It is known that computing the mutual-visibility number of a graph is NP-complete, whereas it has been shown that there are exact formulas for special graph classes like paths, cycles, blocks, cographs, and grids. In this paper, we study the mutual-visibility in distance-hereditary graphs and show that the mutual-visibility number can be computed in linear time for this class.

##### 2.Expansions of averaged truncations of basic hypergeometric series

**Authors:**Michael J. Schlosser, Nian Hong Zhou

**Abstract:** Motivated by recent work of George Andrews and Mircea Merca on the expansion of the quotient of the truncation of Euler's pentagonal number series by the complete series, we provide similar expansion results for averages involving truncations of selected, more general, basic hypergeometric series. In particular, our expansions include new results for averaged truncations of the series appearing in the Jacobi triple product identity, the $q$-Gau{\ss} summation, and the very-well-poised ${}_5\phi_5$ summation. We show how special cases of our expansions can be used to recover various existing results. In addition, we establish new inequalities, such as one for a refinement of the number of partitions into three different colors.

##### 3.Examples and counterexamples in Ehrhart theory

**Authors:**Luis Ferroni, Akihiro Higashitani

**Abstract:** This article provides a comprehensive exposition about inequalities that the coefficients of Ehrhart polynomials and $h^*$-polynomials satisfy under various assumptions. We pay particular attention to the properties of Ehrhart positivity as well as unimodality, log-concavity and real-rootedness for $h^*$-polynomials. We survey inequalities that arise when the polytope has different normality properties. We include statements previously unknown in the Ehrhart theory setting, as well as some original contributions in this topic. We address numerous variations of the conjecture asserting that IDP polytopes have a unimodal $h^*$-polynomial, and construct concrete examples that show that these variations of the conjecture are false. Explicit emphasis is put on polytopes arising within algebraic combinatorics. Furthermore, we describe and construct polytopes having pathological properties on their Ehrhart coefficients and roots, and we indicate for the first time a connection between the notions of Ehrhart positivity and $h^*$-real-rootedness. We investigate the log-concavity of the sequence of evaluations of an Ehrhart polynomial at the non-negative integers. We conjecture that IDP polytopes have a log-concave Ehrhart series. Many additional problems and challenges are proposed.

##### 4.Maximal colourings for graphs

**Authors:**Raffaella Mulas

**Abstract:** We consider two different notions of graph colouring, namely, the $t$-periodic colouring for vertices that has been introduced in 1974 by Bondy and Simonovits, and the periodic colouring for oriented edges that has been recently introduced in the context of spectral theory of non-backtracking operators. For each of these two colourings, we introduce the corresponding colouring number which is given by maximising the possible number of colours. We first investigate these two new colouring numbers individually, and we then show that there is a deep relationship between them.

##### 1.Neighbour-transitive codes in Kneser graphs

**Authors:**Dean Crnković, Daniel R. Hawtin, Nina Mostarac, Andrea Švob

**Abstract:** A code $C$ is a subset of the vertex set of a graph and $C$ is $s$-neighbour-transitive if its automorphism group ${\rm Aut}(C)$ acts transitively on each of the first $s+1$ parts $C_0,C_1,\ldots,C_s$ of the distance partition $\{C=C_0,C_1,\ldots,C_\rho\}$, where $\rho$ is the covering radius of $C$. While codes have traditionally been studied in the Hamming and Johnson graphs, we consider here codes in the Kneser graphs. Let $\Omega$ be the underlying set on which the Kneser graph $K(n,k)$ is defined. Our first main result says that if $C$ is a $2$-neighbour-transitive code in $K(n,k)$ such that $C$ has minimum distance at least $5$, then $n=2k+1$ (i.e., $C$ is a code in an odd graph) and $C$ lies in a particular infinite family or is one particular sporadic example. We then prove several results when $C$ is a neighbour-transitive code in the Kneser graph $K(n,k)$. First, if ${\rm Aut}(C)$ acts intransitively on $\Omega$ we characterise $C$ in terms of certain parameters. We then assume that ${\rm Aut}(C)$ acts transitively on $\Omega$, first proving that if $C$ has minimum distance at least $3$ then either $K(n,k)$ is an odd graph or ${\rm Aut}(C)$ has a $2$-homogeneous (and hence primitive) action on $\Omega$. We then assume that $C$ is a code in an odd graph and ${\rm Aut}(C)$ acts imprimitively on $\Omega$ and characterise $C$ in terms of certain parameters. We give examples in each of these cases and pose several open problems.

##### 2.On rings whose prime ideal sum graphs are line graphs

**Authors:**Praveen Mathil, Jitender Kumar

**Abstract:** Let $R$ be a commutative ring with unity. The prime ideal sum graph of the ring $R$ is the simple undirected graph whose vertex set is the set of all nonzero proper ideals of $R$ and two distinct vertices $I$, $J$ are adjacent if and only if $I + J$ is a prime ideal of $R$. In this paper, we characterize all commutative Artinian rings whose prime ideal sum graphs are line graphs. Finally, we give a description of all commutative Artinian rings whose prime ideal sum graph is the complement of a line graph.

##### 3.A few more Hadamard Partitioned Difference Families

**Authors:**Anamari Nakic

**Abstract:** A $(G,[k_1,\dots,k_t],\lambda)$ {\it partitioned difference family} (PDF) is a partition $\cal B$ of an additive group $G$ into sets ({\it blocks}) of sizes $k_1$, \dots, $k_t$, such that the list of differences of ${\cal B}$ covers exactly $\lambda$ times every non-zero element of $G$. It is called {\it Hadamard} (HPDF) if the order of $G$ is $2\lambda$. The study of HPDFs is motivated by the fact that each of them gives rise, recursively, to infinitely many other PDFs. Apart from the {\it elementary} HPDFs consisting of a Hadamard difference set and its complement, only one HPDF was known. In this article we present three new examples in several groups and we start a general investigation on the possible existence of HPDFs with assigned parameters by means of simple arguments.

##### 4.Schur-Positivity of Short Chords in Matchings

**Authors:**Avichai Marmor

**Abstract:** We prove that the set of matchings with a fixed number of unmatched vertices is Schur-positive with respect to the set of short chords. Two proofs are presented. The first proof applies a new combinatorial criterion for Schur-positivity, while the second is bijective. The coefficients in the Schur expansion are derived, and interpreted in terms of Bessel polynomials. We present a Knuth-like equivalence relation on matchings, and show that every equivalence class corresponds to an irreducible representation. We proceed to find various refined Schur-positive sets, including the set of matchings with a prescribed crossing number and the set of matchings with a given number of pairs of intersecting chords. Finally, we characterize all the matchings $m$ such that the set of matchings avoiding $m$ is Schur-positive.

##### 5.Generalized Pitman-Stanley polytope: vertices and faces

**Authors:**William T. Dugan, Maura Hegarty, Alejandro H. Morales, Annie Raymond

**Abstract:** In 1999, Pitman and Stanley introduced the polytope bearing their name along with a study of its faces, lattice points, and volume. The Pitman-Stanley polytope is well-studied due to its connections to probability, parking functions, the generalized permutahedra, and flow polytopes. Its lattice points correspond to plane partitions of skew shape with entries 0 and 1. Pitman and Stanley remarked that their polytope can be generalized so that lattice points correspond to plane partitions of skew shape with entries $0,1, \ldots , m$. Since then, this generalization has been untouched. We study this generalization and show that it can also be realized as a flow polytope of a grid graph. We give multiple characterizations of its vertices in terms of plane partitions of skew shape and integer flows. For a fixed skew shape, we show that the number of vertices of this polytope is a polynomial in $m$ whose leading term, in certain cases, counts standard Young tableaux of a skew shifted shape. Moreover, we give formulas for the number of faces, as well as generating functions for the number of vertices.

##### 6.Trees with at least $6\ell+11$ vertices are $\ell$-reconstructible

**Authors:**Alexandr V. Kostochka, Mina Nahvi, Douglas B. West, Dara Zirlin

**Abstract:** The $(n-\ell)$-deck of an $n$-vertex graph is the multiset of (unlabeled) subgraphs obtained from it by deleting $\ell$ vertices. An $n$-vertex graph is $\ell$-reconstructible if it is determined by its $(n-\ell)$-deck, meaning that no other graph has the same deck. We prove that every tree with at least $6\ell+11$ vertices is $\ell$-reconstructible.

##### 7.The Containment Game in the plane: between the Firefighter Problem and Conway's Angel Problem

**Authors:**Ohad Noy Feldheim, Itamar Israeli

**Abstract:** The containment game is a full information game for two players, initialised with a set of occupied vertices in an infinite connected graph $G$. On the $t$-th turn, the first player, called Spreader, extends the occupied set to $g(t)$ adjacent vertices, and then the second player, called Container, removes $q$ unoccupied vertices from the graph. If the spreading process continues perpetually -- Spreader wins, and otherwise -- Container wins. For $g=\infty$ this game reduces to a solitaire game for Container, known as the Firefighter Problem. On $\mathbb{Z}^2$, for $q=1/k$ and $g\equiv 1$ it is equivalent to Conway's Angel Problem. We introduce the game, and writing $q(G,g)$ for the set of $q$ values for which Container wins against a given $g(t)$, we study the minimal asymptotics of $g(t)$ such that $q(G,g)=q(G,\infty)$, i.e. for which defeating Spreader is as hard as winning the Firefighter Problem solitaire. We show, by providing explicit winning strategies, a sub-linear upper bound $g(t)=O(t^{6/7})$ and a lower bound of $g(t)=\Omega(t^{1/2})$.

##### 1.Hadamard matrices of orders 60 and 64 with automorphisms of orders 29 and 31

**Authors:**Makoto Araya, Masaaki Harada, Vladimir D. Tonchev

**Abstract:** A classification of Hadamard matrices of order $2p+2$ with an automorphism of order $p$ is given for $p=29$ and $31$. The ternary self-dual codes spanned by the newly found Hadamard matrices of order $60$ with an automorphism of order $29$ are computed, as well as the binary doubly even self-dual codes of length $120$ with generator matrices defined by related Hadamard designs. Several new ternary near-extremal self-dual codes, as well as binary near-extremal doubly even self-dual codes with previously unknown weight enumerators are found.

##### 2.$p$-numerical semigroups of Pell triples

**Authors:**Takao Komatsu, Jiaxin Mu

**Abstract:** For a nonnegative integer $p$, the $p$-numerical semigroup $S_p$ is defined as the set of integers whose nonnegative integral linear combinations of given positive integers $a_1,a_2,\dots,a_\kappa$ with $\gcd(a_1,a_2,\dots,a_\kappa)=1$ are expressed in more than $p$ ways. When $p=0$, $S=S_0$ is the originalnumerical semigroup. The laregest element and the cardinality of $\mathbb N_0\backslash S_p$ are called the $p$-Frobenius number and the $p$-genus, respectively. Their explicit formulas are known for $\kappa=2$, but those for $\kappa\ge 3$ have been found only in some special cases. For some known cases, such as the Fibonacci and the Jacobstal triplets, similar techniques could be applied and explicit formulas such as the $p$-Frobenius number could be found. In this paper, we give explicit formulas for the $p$-Frobenius number and the $p$-genus of Pell numerical semigroups $\bigl(P_i(u),P_{i+2}(u),P_{i+k}(u)\bigr)$. Here, for a given positive integer $u$, Pell-type numbers $P_n(u)$ satisfy the recurrence relation $P_n(u)=u P_{n-1}(u)+P_{n-2}(u)$ ($n\ge 2$) with $P_0(u)=0$ and $P_1(u)=1$. The $p$-Ap\'ery set is used to find the formulas, but it shows a different pattern from those in the known results, and some case by case discussions are necessary.

##### 3.Maximal diameter of integral circulant graphs

**Authors:**Milan Bašić, Aleksandar Ilić, Aleksandar Stamenković

**Abstract:** Integral circulant graphs are proposed as models for quantum spin networks that permit a quantum phenomenon called perfect state transfer. Specifically, it is important to know how far information can potentially be transferred between nodes of the quantum networks modelled by integral circulant graphs and this task is related to calculating the maximal diameter of a graph. The integral circulant graph $ICG_n (D)$ has the vertex set $Z_n = \{0, 1, 2, \ldots, n - 1\}$ and vertices $a$ and $b$ are adjacent if $\gcd(a-b,n)\in D$, where $D \subseteq \{d : d \mid n,\ 1\leq d<n\}$. Motivated by the result on the upper bound of the diameter of $ICG_n(D)$ given in [N. Saxena, S. Severini, I. Shparlinski, \textit{Parameters of integral circulant graphs and periodic quantum dynamics}, International Journal of Quantum Information 5 (2007), 417--430], according to which $2|D|+1$ represents one such bound, in this paper we prove that the maximal value of the diameter of the integral circulant graph $ICG_n(D)$ of a given order $n$ with its prime factorization $p_1^{\alpha_1}\cdots p_k^{\alpha_k}$, is equal to $r(n)$ or $r(n)+1$, where $r(n)=k + |\{ i \ | \alpha_i> 1,\ 1\leq i\leq k \}|$, depending on whether $n\not\in 4N+2$ or not, respectively. Furthermore, we show that, for a given order $n$, a divisor set $D$ with $|D|\leq k$ can always be found such that this bound is attained. Finally, we calculate the maximal diameter in the class of integral circulant graphs of a given order $n$ and cardinality of the divisor set $t\leq k$ and characterize all extremal graphs. We actually show that the maximal diameter can have the values $2t$, $2t+1$, $r(n)$ and $r(n)+1$ depending on the values of $t$ and $n$. This way we further improve the upper bound of Saxena, Severini and Shparlinski and we also characterize all graphs whose diameters are equal to $2|D|+1$, thus generalizing a result in that paper.

##### 4.Universal adjacency spectrum of (proper) power graphs and their complements on some groups

**Authors:**Komal Kumari, Pratima Panigrahi

**Abstract:** The power graph $\mathscr{P}(G)$ of a group $G$ is an undirected graph with all the elements of $G$ as vertices and where any two vertices $u$ and $v$ are adjacent if and only if $u=v^m $ or $v=u^m$, $ m \in$ $\mathbb{Z}$. For a simple graph $H$ with adjacency matrix $A(H)$ and degree diagonal matrix $D(H)$, the universal adjacency matrix is $U(H)= \alpha A(H)+\beta D(H)+ \gamma I +\eta J$, where $\alpha (\neq 0), \beta, \gamma, \eta \in \mathbb{R}$, $I$ is the identity matrix and $J$ is the all-ones matrix of suitable order. One can study many graph-associated matrices, such as adjacency, Laplacian, signless Laplacian, Seidel etc. in a unified manner through the universal adjacency matrix of a graph. Here we study universal adjacency eigenvalues and eigenvectors of power graphs, proper power graphs and their complements on the group $\mathbb{Z}_n$, dihedral group ${D}_n$, and the generalized quaternion group ${Q}_n$. Spectral results of no kind for the complement of power graph on any group were obtained before. We determine the full spectrum in some particular cases. Moreover, several existing results can be obtained as very specific cases of some results of the paper.

##### 5.MaxCut in graphs with sparse neighborhoods

**Authors:**Jinghua Deng, Jianfeng Hou, Siwei Lin, Qinghou Zeng

**Abstract:** Let $G$ be a graph with $m$ edges and let $\mathrm{mc}(G)$ denote the size of a largest cut of $G$. The difference $\mathrm{mc}(G)-m/2$ is called the surplus $\mathrm{sp}(G)$ of $G$. A fundamental problem in MaxCut is to determine $\mathrm{sp}(G)$ for $G$ without specific structure, and the degree sequence $d_1,\ldots,d_n$ of $G$ plays a key role in getting the lower bound of $\mathrm{sp}(G)$. A classical example, given by Shearer, is that $\mathrm{sp}(G)=\Omega(\sum_{i=1}^n\sqrt d_i)$ for triangle-free graphs $G$, implying that $\mathrm{sp}(G)=\Omega(m^{3/4})$. It was extended to graphs with sparse neighborhoods by Alon, Krivelevich and Sudakov. In this paper, we establish a novel and stronger result for a more general family of graphs with sparse neighborhoods. Our result can derive many well-known bounds on $\mathrm{sp}(G)$ in $H$-free graphs $G$ for different $H$, such as the triangle, the even cycle, the graphs having a vertex whose removal makes the graph acyclic, or the complete bipartite graph $K_{s,t}$ with $s\in \{2,3\}$. It can also deduce many new (tight) bounds on $\mathrm{sp}(G)$ in $H$-free graphs $G$ when $H$ is any graph having a vertex whose removal results in a bipartite graph with relatively small Tur\'{a}n number, especially the even wheel. This contributes to a conjecture raised by Alon, Krivelevich and Sudakov. Moreover, we give a new family of graphs $H$ such that $\mathrm{sp}(G)=\Omega(m^{3/4+\epsilon(H)})$ for some constant $\epsilon(H)>0$ in $H$-free graphs $G$, giving an evidence to a conjecture suggested by Alon, Bollob\'as, Krivelevich and Sudakov.

##### 6.Some results on extremal spectral radius of hypergraph

**Authors:**Guanglong Yu

**Abstract:** For a $hypergraph$ $\mathcal{G}=(V, E)$ with a nonempty vertex set $V=V(\mathcal{G})$ and an edge set $E=E(\mathcal{G})$, its $adjacency$ $matrix$ $\mathcal {A}_{\mathcal{G}}=[(\mathcal {A}_{\mathcal{G}})_{ij}]$ is defined as $(\mathcal {A}_{\mathcal{G}})_{ij}=\sum_{e\in E_{ij}}\frac{1}{|e| - 1}$, where $E_{ij} = \{e\in E\, |\, i, j \in e\}$. The $spectral$ $radius$ of a hypergraph $\mathcal{G}$, denoted by $\rho(\mathcal {G})$, is the maximum modulus among all eigenvalues of $\mathcal {A}_{\mathcal{G}}$. In this paper, we get a formula about the spectral radius which link the ordinary graph and the hypergraph, and represent some results on the spectral radius changing under some graphic structural perturbations. Among all $k$-uniform ($k\geq 3$) unicyclic hypergraphs with fixed number of vertices, the hypergraphs with the minimum, the second the minimum spectral radius are completely determined, respectively; among all $k$-uniform ($k\geq 3$) unicyclic hypergraphs with fixed number of vertices and fixed girth, the hypergraphs with the maximum spectral radius are completely determined; among all $k$-uniform ($k\geq 3$) $octopuslike$ hypergraphs with fixed number of vertices, the hypergraphs with the minimum spectral radius are completely determined. As well, for $k$-uniform ($k\geq 3$) $lollipop$ hypergraphs, we get that the spectral radius decreases with the girth increasing.

##### 7.Algorithms and hardness for Metric Dimension on digraphs

**Authors:**Antoine Dailly, Florent Foucaud, Anni Hakanen

**Abstract:** In the Metric Dimension problem, one asks for a minimum-size set R of vertices such that for any pair of vertices of the graph, there is a vertex from R whose two distances to the vertices of the pair are distinct. This problem has mainly been studied on undirected graphs and has gained a lot of attention in the recent years. We focus on directed graphs, and show how to solve the problem in linear-time on digraphs whose underlying undirected graph (ignoring multiple edges) is a tree. This (nontrivially) extends a previous algorithm for oriented trees. We then extend the method to unicyclic digraphs (understood as the digraphs whose underlying undirected multigraph has a unique cycle). We also give a fixed-parameter-tractable algorithm for digraphs when parameterized by the directed modular-width, extending a known result for undirected graphs. Finally, we show that Metric Dimension is NP-hard even on planar triangle-free acyclic digraphs of maximum degree 6.

##### 8.Local central limit theorem for triangle counts in sparse random graphs

**Authors:**Pedro Araújo, Letícia Mattos

**Abstract:** Let $X_H$ be the number of copies of a fixed graph $H$ in $G(n,p)$. In 2016, Gilmer and Kopparty conjectured that a local central limit theorem should hold for $X_H$ as long as $H$ is connected, $p \gg n^{-1/m(H)}$ and $n^2(1-p)\gg 1$. Recently, Sah and Sahwney showed that the Gilmer--Kopparty conjecture holds for constant $p$. In this paper, we show that the Gilmer--Kopparty conjecture holds for triangle counts in the sparse range. More precisely, there exists $C>0$ such that if $p \in (Cn^{-1/2}, 1/2)$, then \[ \sup_{x\in \mathcal{L}}\left| \dfrac{1}{\sqrt{2\pi}}e^{-x^2/2}-\sigma\cdot \mathbb{P}(X^* = x)\right|\rightarrow 0,\] where $\sigma^2 = \mathbb{V}\text{ar}(X_{K_3})$, $X^{*}=(X_{K_3}-\mathbb{E}(X_{K_3}))/\sigma$ and $\mathcal{L}$ is the support of $X^*$. By combining our result with the results of R\"ollin--Ross and Gilmer--Kopparty, this establishes the Gilmer--Kopparty conjecture for triangle counts for $n^{-1}\ll p < c$, for any constant $c\in (0,1)$. Our result is the first local central limit theorem for subgraph counts above the $m_2$-density.

##### 9.On colouring oriented graphs of large girth

**Authors:**P. Mark Kayll, Michael Morris

**Abstract:** We prove that for every oriented graph $D$ and every choice of positive integers $k$ and $\ell$, there exists an oriented graph $D^*$ along with a surjective homomorphism $\psi\colon V(D^*) \to V(D)$ such that: (i) girth$(D^*) \geq\ell$; (ii) for every oriented graph $C$ with at most $k$ vertices, there exists a homomorphism from $D^*$ to $C$ if and only if there exists a homomorphism from $D$ to $C$; and (iii) for every $D$-pointed oriented graph $C$ with at most $k$ vertices and for every homomorphism $\varphi\colon V(D^*) \to V(C)$ there exists a unique homomorphism $f\colon V(D) \to V(C)$ such that $\varphi=f \circ \psi$. Determining the oriented chromatic number of an oriented graph $D$ is equivalent to finding the smallest integer $k$ such that $D$ admits a homomorphism to an order-$k$ tournament, so our main theorem yields results on the girth and oriented chromatic number of oriented graphs. While our main proof is probabilistic (hence nonconstructive), for any given $\ell\geq 3$ and $k\geq 5$, we include a construction of an oriented graph with girth $\ell$ and oriented chromatic number $k$.

##### 1.Phase Transitions of Structured Codes of Graphs

**Authors:**Bo Bai, Yu Gao, Jie Ma, Yuze Wu

**Abstract:** We consider the symmetric difference of two graphs on the same vertex set $[n]$, which is the graph on $[n]$ whose edge set consists of all edges that belong to exactly one of the two graphs. Let $\mathcal{F}$ be a class of graphs, and let $M_{\mathcal{F}}(n)$ denote the maximum possible cardinality of a family $\mathcal{G}$ of graphs on $[n]$ such that the symmetric difference of any two members in $\mathcal{G}$ belongs to $\mathcal{F}$. These concepts are recently investigated by Alon, Gujgiczer, K\"{o}rner, Milojevi\'{c}, and Simonyi, with the aim of providing a new graphic approach to coding theory. In particular, $M_{\mathcal{F}}(n)$ denotes the maximum possible size of this code. Existing results show that as the graph class $\mathcal{F}$ changes, $M_{\mathcal{F}}(n)$ can vary from $n$ to $2^{(1+o(1))\binom{n}{2}}$. We study several phase transition problems related to $M_{\mathcal{F}}(n)$ in general settings and present a partial solution to a recent problem posed by Alon et. al.

##### 2.Components of domino tilings under flips in toroidal grids

**Authors:**Qianqian Liu, Yaxian Zhang, Heping Zhang

**Abstract:** In a region $R$ consisting of unit squares, a (domino) tiling is a collection of dominoes (the union of two adjacent squares) which pave fully the region. The flip graph of $R$ is defined on the set of all tilings of $R$ where two tilings are adjacent if we change one from the other by a flip (a $90^{\circ}$ rotation of a pair of side-by-side dominoes). Let $n\geq 1$ and $m\geq 2$ be integers. In a recent paper it was proved that the flip graph of $(2n+1)\times 2m$ quadriculated torus consists of two isomorphic components. In this paper, we generalize the result to the toroidal grid $T(2n+1,2m,2r)$ which is obtained from an $(2n+1)\times 2m$ chessboard by sticking the left and right sides and then identifying the top and bottom sides with a torsion $2r$ squares for $1\leq r\leq m$. For a tiling $t$, we associate an integer called forcing number as the minimum number of dominoes in $t$ that is contained in no other tiling. As an application, we obtain that the forcing numbers of all tilings of $T(2n+1,2m,2r)$ form an integer-interval. Moreover, we prove that the maximum value of the interval is $\frac{m(2n+1)}{2}$ if $\frac{m}{(r,m)}$ is even, and $\frac{m(2n+1)+(r,m)}{2}$ otherwise.

##### 3.Induced $C_4$-free subgraphs with high average degree

**Authors:**Xiying Du, António Girão, Zach Hunter, Rose McCarty, Alex Scott

**Abstract:** We prove that there exists a constant $C$ so that for all $s,k \in \mathbb{N}$, every $K_{s,s}$-free graph with average degree at least $k^{Cs^3}$ contains an induced subgraph which is $C_4$-free and has average degree at least $k$. It was known that some function of $s$ and $k$ suffices, but this is the first explicit bound. Using this theorem, we give short and streamlined proofs of the following three corollaries. First, we show that there exists a constant $C$ so that for all $k \in \mathbb{N}$, every graph with average degree at least $k^{Ck^6}$ contains a bipartite subgraph (not necessarily induced) which is $C_4$-free and has average degree at least $k$. This almost matches the recent bound of $k^{Ck^2}$ due to Montgomery, Pokrovskiy, and Sudakov. Next, we show that there exists a constant $C$ so that for all $s,k \in \mathbb{N}$, every $K_{s,s}$-free graph with average degree at least $k^{Cs^3}$ contains an induced subdivision of $K_k$. This is the first quantitative improvement on a well-known theorem of K\"uhn and Osthus; their proof gives a bound that is triply exponential in both $k$ and $s$. Finally, we show that for any hereditary degree-bounded class $\mathcal{F}$, there exists a constant $C_\mathcal{F}$ so that $(C_\mathcal{F})^{s^3}$ is a degree-bounding function for $\mathcal{F}$. This is the first bound of any type on the rate of growth of such functions. It is open whether there is always a polynomial degree-bounding function.

##### 4.The skeleton of a convex polytope

**Authors:**Takayuki Hibi, Aki Mori

**Abstract:** Let ${\rm sk}({\mathcal P})$ denote the $1$-skeleton of an convex polytope ${\mathcal P}$. Let $C$ be a clique (=complete subgraph) of ${\rm sk}({\mathcal P})$ and ${\rm conv}(C)$ the convex hull of the vertices of ${\mathcal P}$ belonging to $C$. In general, ${\rm conv}(C)$ may not be a face of ${\mathcal P}$. It will be proved that ${\rm conv}(C)$ is a face of ${\mathcal P}$ if ${\mathcal P}$ is either the order polytope ${\mathcal O}(P)$ of a finite partially ordered set $P$ or the stable set polytope ${\rm Stab}(G)$ of a finite simple graph $G$. In other words, when ${\mathcal P}$ is either ${\mathcal O}(P)$ or ${\rm Stab}(G)$, the simplicial complex consisting of simplices which are faces of ${\mathcal P}$ is the clique complex of ${\rm sk}({\mathcal P})$.

##### 5.A structural duality for path-decompositions into parts of small radius

**Authors:**Sandra Albrechtsen, Reinhard Diestel, Ann-Kathrin Elm, Eva Fluck, Raphael W. Jacobs, Paul Knappe, Paul Wollan

**Abstract:** Given an arbitrary class $\mathcal{H}$ of graphs, we investigate which graphs admit a decomposition modelled on a graph in $\mathcal{H}$ into parts of small radius. The $\mathcal{H}$-decompositions that we consider here generalise the notion of tree-decompositions. We identify obstructions to such $\mathcal{H}$-decompositions of small radial width, and we prove that these obstructions occur in every graph of sufficiently large radial $\mathcal{H}$-width for the classes $\mathcal{H}$ of paths, of cycles and of subdivided stars.

##### 6.Toggling, rowmotion, and homomesy on interval-closed sets

**Authors:**Jennifer Elder, Nadia Lafrenière, Erin McNicholas, Jessica Striker, Amanda Welch

**Abstract:** Interval-closed sets of a poset are a natural superset of order ideals. We initiate the study of interval-closed sets of finite posets from enumerative and dynamical perspectives. Following [Striker, 2018], we study the generalized toggle group of interval-closed sets. In particular, we define rowmotion on interval-closed sets as a product of these toggles. Our main general theorem is an intricate global characterization of rowmotion on interval-closed sets, which we show is equivalent to the toggling definition. We also study specific posets; we enumerate interval-closed sets of ordinal sums of antichains, completely describe their rowmotion orbits, and prove a homomesy result involving the signed cardinality statistic. Finally, we study interval-closed sets of product of chains posets, proving further results about enumeration and homomesy.

##### 7.Colouring Digraphs

**Authors:**Guillaume Aubian

**Abstract:** The aim of this thesis is to investigate how the structure of a digraph affects its dichromatic number and to extend various results on undirected colouring to digraphs. In the first part of this thesis, we examine how the dichromatic number interacts with other metrics. First, we consider the degree, which is the maximum number of neighbours of a vertex. In the undirected case, this corresponds to Brooks' theorem, a celebrated theorem with multiple variations and generalizations. In the directed case, there is no natural metric corresponding to the maximum degree, so we explore how different notions of maximum directed degree lead to either Brooks-like theorems or impossibility results. We also investigate the maximum local-arc connectivity, a metric that encompasses several degree-like metrics. The second part of this manuscript focuses on a directed analogue of the Gy\'arf\'as-Sumner conjecture. The Gy\'arf\'as-Sumner conjecture tries to characterize sets S of undirected graphs such that graphs with large enough chromatic number must contain a graph of S. This conjecture is still largely open. On digraphs, a corresponding conjecture was proposed by Aboulker, Charbit, and Naserasr. We prove several subcases of this conjecture, mainly demonstrating that certain classes of digraphs have bounded dichromatic number. In the last part of this thesis, we address the d-edge-defective-colouring problem, which involves colouring edges of a multigraph such that, for any vertex, no colour appears on more than d of its incident edges. When d equals one, this corresponds to the infamous edge-colouring problem. Shannon established a tight bound on the number of colours needed relative to the maximum degree when d equals one, and we extend this result to any value of d. We also explore this problem on simple graphs and prove results that extend Vizing's theorem to any value of d.

##### 8.Ramsey numbers and the Zarankiewicz problem

**Authors:**David Conlon, Sam Mattheus, Dhruv Mubayi, Jacques Verstraëte

**Abstract:** Building on recent work of Mattheus and Verstra\"ete, we establish a general connection between Ramsey numbers of the form $r(F,t)$ for $F$ a fixed graph and a variant of the Zarankiewicz problem asking for the maximum number of 1s in an $m$ by $n$ $0/1$-matrix that does not have any matrix from a fixed finite family $\mathcal{L}(F)$ derived from $F$ as a submatrix. As an application, we give new lower bounds for the Ramsey numbers $r(C_5,t)$ and $r(C_7,t)$, namely, $r(C_5,t) = \tilde\Omega(t^{\frac{10}{7}})$ and $r(C_7,t) = \tilde\Omega(t^{\frac{5}{4}})$. We also show how the truth of a plausible conjecture about Zarankiewicz numbers would allow an approximate determination of $r(C_{2\ell+1}, t)$ for any fixed integer $\ell \geq 2$.

##### 1.On arithmetic sums of Cantor-type sequences of integers

**Authors:**Norbert Hegyvári

**Abstract:** We are looking for integer sets that resemble classical Cantor set and investigate the structure of their sum sets. Especially we investigate $FS(B)$ the subset sum of sequence type $B=\{\lfloor p^n\alpha\rfloor\}^\infty_{n=0}$. When $p=2$, then we prove $FS(B)+FS(B)=\N$ by analogy with the Cantor set, and some structure theorem for $p>2$

##### 2.Universal lower bound for community structure of sparse graphs

**Authors:**Vilhelm Agdur, Nina Kamčev, Fiona Skerman

**Abstract:** We prove new lower bounds on the modularity of graphs. Specifically, the modularity of a graph $G$ with average degree $\bar d$ is $\Omega(\bar{d}^{-1/2})$, under some mild assumptions on the degree sequence of $G$. The lower bound $\Omega(\bar{d}^{-1/2})$ applies, for instance, to graphs with a power-law degree sequence or a near-regular degree sequence. It has been suggested that the relatively high modularity of the Erd\H{o}s-R\'enyi random graph $G_{n,p}$ stems from the random fluctuations in its edge distribution, however our results imply high modularity for any graph with a degree sequence matching that typically found in $G_{n,p}$. The proof of the new lower bound relies on certain weight-balanced bisections with few cross-edges, which build on ideas of Alon [Combinatorics, Probability and Computing (1997)] and may be of independent interest.

##### 3.Integral Laplacian graphs with a unique double Laplacian eigenvalue, II

**Authors:**Abdul Hameed, Mikhail Tyaglov

**Abstract:** The set $S_{\{i,j\}_{n}^{m}}=\{0,1,2,\ldots,m-1,m,m,m+1,\ldots,n-1,n\}\setminus\{i,j\},\quad 0<i<j\leqslant n$, is called Laplacian realizable if there exists a simple connected graph $G$ whose Laplacian spectrum is $S_{\{i,j\}_{n}^{m}}$. In this case, the graph $G$ is said to realize $S_{\{i,j\}_{n}^{m}}$. In this paper, we completely describe graphs realizing the sets $S_{\{i,j\}_{n}^{m}}$ with $m=1,2$ and determine the structure of these graphs.

##### 4.Overlaps in Field Generated Circular Planar Nearrings

**Authors:**Wen-Fong Ke, Hubert Kiechle

**Abstract:** We investigate circular planar nearrings constructed from finite fields as well the complex number field using a multiplicative subgroup of order $k$, and characterize the overlaps of the basic graphs which arise in the associated $2$-designs.

##### 5.Attainable bounds for algebraic connectivity and maximally-connected regular graphs

**Authors:**Geoffrey Exoo, Theodore Kolokolnikov, Jeanette Janssen, Timothy Salamon

**Abstract:** We derive attainable upper bounds on the algebraic connectivity (spectral gap) of a regular graph in terms of its diameter and girth. This bound agrees with the well-known Alon-Boppana-Friedman bound for graphs of even diameter, but is an improvement for graphs of odd diameter. For the girth bound, we show that only Moore graphs can attain it, and these only exist for very few possible girths. For diameter bound, we use a combination of stochastic algorithms and exhaustive search to find graphs which attain it. For 3-regular graphs, we find attainable graphs for all diameters $D$ up to and including $D=9$ (the case of $D=10$ is open). These graphs are extremely rare and also have high girth; for example we found exactly 45 distinct cubic graphs on 44 vertices attaining the upper bound when $D=7$; all have girth 8 (out of a total of about $10^{20}$ cubic graphs on 44 vertices, including 266362 having girth 8). We also exhibit families of $d$-regular graphs attaining upper bounds with $D=3$ and $4$, and with $g=6.$ Several conjectures are proposed.

##### 6.Minimum $k$-critical-bipartite graphs: the irregular Case

**Authors:**Sylwia Cichacz, Agieszka Görlich, Karol Suchan

**Abstract:** We study the problem of finding a minimum $k$-critical-bipartite graph of order $(n,m)$: a bipartite graph $G=(U,V;E)$, with $|U|=n$, $|V|=m$, and $n>m>1$, which is $k$-critical-bipartite, and the tuple $(|E|, \Delta_U, \Delta_V)$, where $\Delta_U$ and $\Delta_V$ denote the maximum degree in $U$ and $V$, respectively, is lexicographically minimum over all such graphs. $G$ is $k$-critical-bipartite if deleting at most $k=n-m$ vertices from $U$ yields $G'$ that has a complete matching, i.e., a matching of size $m$. Cichacz and Suchan solved the problem for biregular bipartite graphs. Here, we extend their results to bipartite graphs that are not biregular. We also prove tight lower bounds on the connectivity of $k$-critical-bipartite graphs.

##### 7.Stability properties of inner plethyms (Lecture Notes)

**Authors:**Jean-Yves Thibon

**Abstract:** The inner plethysm of symmetric functions corresponds to the $\lambda$-ring operations of the representation ring $R({\mathfrak S}_n)$ of the symmetric group. It is known since the work of Littlewood that this operation possesses stability properties w.r.t. $n$. These properties have been explained in terms of vertex operators [Scharf and Thibon, Adv. Math. 104 (1994), 30-58]. Another approach [Orellana and Zabrocki, Adv. Math. 390 (2021), \# 107943], based on an expression of character values as symmetric functions of the eigenvalues of permutation matrices, has been proposed recently. This note develops the theory from scratch, discusses the link between both approaches and provides new proofs of some recent results.

##### 8.Planar algebras for the Young graph and the Khovanov Heisenberg category

**Authors:**Shinji Koshida

**Abstract:** This paper studies planar algebras of Jones' style associated with the Young graph. We first see that, every time we fix a harmonic function on the Young graph, we may obtain a planar algebra whose structure is defined in terms of a state sum over the ways of filling planar tangles with Young diagrams. We delve into the case that the harmonic function is related to the Plancherel measures on Young diagrams. Along with an element that is depicted as a cross of two strings, we see that the defining relations among morphisms for the Khovanov Heisenberg category are recovered in the planar algebra. We also identify certain elements in the planar algebra with particular functions of Young diagrams that include the moments, Boolean cumulants and normalized characters. This paper thereby bridges diagramatical categorification and asymptotic representation theory. In fact, the Khovanov Heisenberg category is one of the most fundamental examples of diagramatical categorification whereas the harmonic functions on the Young graph have been a central object in the asymptotic representation theory of symmetric groups.

##### 9.Whitney Twins, Whitney Duals, and Operadic Partition Posets

**Authors:**Rafael S. González D'León, Joshua Hallam, Yeison A. Quiceno D

**Abstract:** We say that a pair of nonnegative integer sequences $(\{a_k\}_{k\geq 0},\{b_k\}_{k\geq 0})$ is Whitney-realizable if there exists a poset $P$ for which (the absolute values) of the Whitney numbers of the first and second kind are given by the numbers $a_k$ and $b_k$ respectively. The pair is said to be Whitney-dualizable if, in addition, there exists another poset $Q$ for which their Whitney numbers of the first and second kind are instead given by $b_k$ and $a_k$ respectively. In this case, we say that $P$ and $Q$ are Whitney duals. We use results on Whitney duality, recently developed by the first two authors, to exhibit a family of sequences which allows for multiple realizations and Whitney-dual realizations. More precisely, we study edge labelings for the families of posets of pointed partitions $\Pi_n^{\bullet}$ and weighted partitions $\Pi_n^{w}$ which are associated to the operads $\mathcal{P}erm$ and $\mathcal{C}om^2$ respectively. The first author and Wachs proved that these two families of posets share the same pair of Whitney numbers. We find EW-labelings for $\Pi_n^{\bullet}$ and $\Pi_n^{w}$ and use them to show that they also share multiple nonisomorphic Whitney dual posets. In addition to EW-labelings, we also find two new EL-labelings for $\Pi_n^\bullet$ answering a question of Chapoton and Vallette. Using these EL-labelings of $\Pi_n^\bullet$, and an EL-labeling of $\Pi_n^w$ introduced by the first author and Wachs, we give combinatorial descriptions of bases for the operads $\mathcal{P}re\mathcal{L}ie, \mathcal{P}erm,$ and $\mathcal{C}om^2$. We also show that the bases for $\mathcal{P}erm$ and $\mathcal{C}om^2$ are PBW bases.

##### 1.Bootstrap percolation in strong products of graphs

**Authors:**Boštjan Brešar, Jaka Hedžet

**Abstract:** Given a graph $G$ and assuming that some vertices of $G$ are infected, the $r$-neighbor bootstrap percolation rule makes an uninfected vertex $v$ infected if $v$ has at least $r$ infected neighbors. The $r$-percolation number, $m(G,r)$, of $G$ is the minimum cardinality of a set of initially infected vertices in $G$ such that after continuously performing the $r$-neighbor bootstrap percolation rule each vertex of $G$ eventually becomes infected. In this paper, we consider percolation numbers of strong products of graphs. If $G$ is the strong product $G_1\boxtimes \cdots \boxtimes G_k$ of $k$ connected graphs, we prove that $m(G,r)=r$ as soon as $r\le 2^{k-1}$ and $|V(G)|\ge r$. As a dichotomy, we present a family of strong products of $k$ connected graphs with the $(2^{k-1}+1)$-percolation number arbitrarily large. We refine these results for strong products of graphs in which at least two factors have at least three vertices. In addition, when all factors $G_i$ have at least three vertices we prove that $m(G_1 \boxtimes \dots \boxtimes G_k,r)\leq 3^{k-1} -k$ for all $r\leq 2^k-1$, and we again get a dichotomy, since there exist families of strong products of $k$ graphs such that their $2^{k}$-percolation numbers are arbitrarily large. While $m(G\boxtimes H,3)=3$ if both $G$ and $H$ have at least three vertices, we also characterize the strong prisms $G\boxtimes K_2$ for which this equality holds. Some of the results naturally extend to infinite graphs, and we briefly consider percolation numbers of strong products of two-way infinite paths.

##### 2.Constructive proof of the cycle double cover conjecture

**Authors:**Jens Walter Fischer

**Abstract:** Based on a construction iterating the line graph operator twice, the cycle double cover conjecture is proven in a constructive way using a perspective inspired by statistical mechanics and spin systems. In the resulting graph, deletion of specific subgraphs gives a union of cycles which via projection to the underlying graph results in a double cycle cover.

##### 3.The Frobenius transform of a symmetric function

**Authors:**Mitchell Lee

**Abstract:** We define an abelian group homomorphism $\mathscr{F}$, which we call the Frobenius transform, from the ring of symmetric functions to the ring of the symmetric power series. The matrix entries of $\mathscr{F}$ in the Schur basis are the restriction coefficients $r_\lambda^\mu = \dim \operatorname{Hom}_{\mathfrak{S}_n}(V_\mu, \mathbb{S}^\lambda \mathbb{C}^n)$, which are known to be nonnegative integers but have no known combinatorial interpretation. The Frobenius transform satisfies the identity $\mathscr{F}\{fg\} = \mathscr{F}\{f\} \ast \mathscr{F}\{g\}$, where $\ast$ is the Kronecker product. We prove for all symmetric functions $f$ that $\mathscr{F}\{f\} = \mathscr{F}_{\mathrm{Sur}}\{f\} \cdot (1 + h_1 + h_2 + \cdots)$, where $\mathscr{F}_{\mathrm{Sur}}\{f\}$ is a symmetric function with the same degree and leading term as $f$. Then, we compute the matrix entries of $\mathscr{F}_{\mathrm{Sur}}\{f\}$ and $\mathscr{F}^{-1}_{\mathrm{Sur}}\{f\}$ in the complete homogeneous, elementary, and power sum symmetric function bases, giving combinatorial interpretations of the coefficients where possible. In particular, the matrix entries of $\mathscr{F}^{-1}_{\mathrm{Sur}}\{f\}$ in the elementary basis count words with a constraint on their Lyndon factorization. As an example application of our main results, we prove that $r_\lambda^\mu = 0$ if the Young diagram of $\mu$ contains a square of side length greater than $2^{\lambda_1 - 1}$, and this inequality is tight.

##### 4.Redicolouring digraphs: directed treewidth and cycle-degeneracy

**Authors:**Nicolas Nisse, Lucas Picasarri-Arrieta, Ignasi Sau

**Abstract:** Given a digraph $D=(V,A)$ on $n$ vertices and a vertex $v\in V$, the cycle-degree of $v$ is the minimum size of a set $S \subseteq V(D) \setminus \{v\}$ intersecting every directed cycle of $D$ containing $v$. From this definition of cycle-degree, we define the $c$-degeneracy (or cycle-degeneracy) of $D$, which we denote by $\delta^*_c(D)$. It appears to be a nice generalisation of the undirected degeneracy. In this work, using this new definition of cycle-degeneracy, we extend several evidences for Cereceda's conjecture to digraphs. The $k$-dicolouring graph of $D$, denoted by $\mathcal{D}_k(D)$, is the undirected graph whose vertices are the $k$-dicolourings of $D$ and in which two $k$-dicolourings are adjacent if they differ on the colour of exactly one vertex. We show that $\mathcal{D}_k(D)$ has diameter at most $O_{\delta^*_c(D)}(n^{\delta^*_c(D) + 1})$ (respectively $O(n^2)$ and $(\delta^*_c(D)+1)$) when $k$ is at least $\delta^*_c(D)+2$ (respectively $\frac{3}{2}(\delta^*_c(D)+1)$ and $2(\delta^*_c(D)+1)$). This improves known results on digraph redicolouring (Bousquet et al.). Next, we extend a result due to Feghali to digraphs, showing that $\mathcal{D}_{d+1}(D)$ has diameter at most $O_{d,\epsilon}(n(\log n)^{d-1})$ when $D$ has maximum average cycle-degree at most $d-\epsilon$. We then show that two proofs of Bonamy and Bousquet for undirected graphs can be extended to digraphs. The first one uses the digrundy number of a digraph and the second one uses the $\mathscr{D}$-width. Finally, we give a general theorem which makes a connection between the recolourability of a digraph $D$ and the recolourability of its underlying graph $UG(D)$. This result directly extends a number of results on planar graph recolouring to planar digraph redicolouring.

##### 5.Notes for Neighborly Partitions

**Authors:**Kathleen O'Hara, Dennis Stanton

**Abstract:** A proof of the first Rogers-Ramanujan identity is given using admissible neighborly partitions. This completes a program initiated by Mohsen and Mourtada. The admissible neighborly partitions involve an unusual mod 3 condition on the parts.

##### 6.The sum-product problem for small sets

**Authors:**Ginny Ray Clevenger, Haley Havard, Patch Heard, Andrew Lott, Alex Rice, Brittany Wilson

**Abstract:** For $A\subseteq \mathbb{R}$, let $A+A=\{a+b: a,b\in A\}$ and $AA=\{ab: a,b\in A\}$. For $k\in \mathbb{N}$, let $SP(k)$ denote the minimum value of $\max\{|A+A|, |AA|\}$ over all $A\subseteq \mathbb{N}$ with $|A|=k$. Here we establish $SP(k)=3k-3$ for $2\leq k \leq 7$, the $k=7$ case achieved for example by $\{1,2,3,4,6,8,12\}$, while $SP(k)=3k-2$ for $k=8,9$, the $k=9$ case achieved for example by $\{1,2,3,4,6,8,9,12,16\}$. For $4\leq k \leq 7$, we provide two proofs using different applications of Freiman's $3k-4$ theorem; one of the proofs includes extensive case analysis on the product sets of $k$-element subsets of $(2k-3)$-term arithmetic progressions. For $k=8,9$, we apply Freiman's $3k-3$ theorem for product sets, and investigate the sumset of the union of two geometric progressions with the same common ratio $r>1$, with separate treatments of the overlapping cases $r\neq 2$ and $r\geq 2$.

##### 7.The planar Turán number of the seven-cycle

**Authors:**Ervin Győri, Alan Li, Runtian Zhou

**Abstract:** The planar Tur\'an number, $ex_\mathcal{P}(n,H)$, is the maximum number of edges in an $n$-vertex planar graph which does not contain $H$ as a subgraph. The topic of extremal planar graphs was initiated by Dowden (2016). He obtained sharp upper bound for both $ex_\mathcal{P}(n,C_4)$ and $ex_\mathcal{P}(n,C_5)$. Later on, D. Ghosh et al. obtained sharp upper bound of $ex_\mathcal{P}(n,C_6)$ and proposed a conjecture on $ex_\mathcal{P}(n,C_k)$ for $k\geq 7$. In this paper, we give a sharp upper bound $ex_\mathcal{P}(n,C_7)\leq {18\over 7}n-{48\over 7}$, which satisfies the conjecture of D. Ghosh et al. It turns out that this upper bound is also sharp for $ex_\mathcal{P}(n,\{K_4,C_7\})$, the maximum number of edges in an $n$-vertex planar graph which does not contain $K_4$ or $C_7$ as a subgraph.

##### 8.Uniform sets with few progressions via colorings

**Authors:**Mingyang Deng, Jonathan Tidor, Yufei Zhao

**Abstract:** Ruzsa asked whether there exist Fourier-uniform subsets of $\mathbb Z/N\mathbb Z$ with density $\alpha$ and 4-term arithmetic progression (4-APs) density at most $\alpha^C$, for arbitrarily large $C$. Gowers constructed Fourier uniform sets with density $\alpha$ and 4-AP density at most $\alpha^{4+c}$ for some small constant $c>0$. We show that an affirmative answer to Ruzsa's question would follow from the existence of an $N^{o(1)}$-coloring of $[N]$ without symmetrically colored 4-APs. For a broad and natural class of constructions of Fourier-uniform subsets of $\mathbb Z/N\mathbb Z$, we show that Ruzsa's question is equivalent to our arithmetic Ramsey question. We prove analogous results for all even-length APs. For each odd $k\geq 5$, we show that there exist $U^{k-2}$-uniform subsets of $\mathbb Z/N\mathbb Z$ with density $\alpha$ and $k$-AP density at most $\alpha^{c_k \log(1/\alpha)}$. We also prove generalizations to arbitrary one-dimensional patterns.

##### 1.Hochschild polytopes

**Authors:**Vincent Pilaud, Daria Poliakova

**Abstract:** The $(m,n)$-multiplihedron is a polytope whose faces correspond to $m$-painted $n$-trees, and whose oriented skeleton is the Hasse diagram of the rotation lattice on binary $m$-painted $n$-trees. Deleting certain inequalities from the facet description of the $(m,n)$-multiplihedron, we construct the $(m,n)$-Hochschild polytope whose faces correspond to $m$-lighted $n$-shades, and whose oriented skeleton is the Hasse diagram of the rotation lattice on unary $m$-lighted $n$-shades. Moreover, there is a natural shadow map from $m$-painted $n$-trees to $m$-lighted $n$-shades, which turns out to define a meet semilattice morphism of rotation lattices. In particular, when $m=1$, our Hochschild polytope is a deformed permutahedron whose oriented skeleton is the Hasse diagram of the Hochschild lattice.

##### 2.Projective dimension of weakly chordal graphic arrangements

**Authors:**Takuro Abe, Lukas Kühne, Paul Mücksch, Leonie Mühlherr

**Abstract:** A graphic arrangement is a subarrangement of the braid arrangement whose set of hyperplanes is determined by an undirected graph. A classical result due to Stanley, Edelman and Reiner states that a graphic arrangement is free if and only if the corresponding graph is chordal, i.e., the graph has no chordless cycle with four or more vertices. In this article we extend this result by proving that the module of logarithmic derivations of a graphic arrangement has projective dimension at most one if and only if the corresponding graph is weakly chordal, i.e., the graph and its complement have no chordless cycle with five or more vertices.

##### 3.Asymmetry of 2-step Transit Probabilities in 2-Coloured Regular Graphs

**Authors:**Ron Gray, J. Robert Johnson

**Abstract:** Suppose that the vertices of a regular graph are coloured red and blue with an equal number of each (we call this a balanced colouring). Since the graph is undirected, the number of edges from a red vertex to a blue vertex is clearly the same as the number of edges from a blue vertex to a red vertex. However, if instead of edges we count walks of length $2$, then this symmetry disappears. Our aim in this paper is to investigate how extreme this asymmetry can be. Our main question is: Given a $d$-regular graph, for which pairs $(x,y)\in[0,1]^2$ is there a balanced colouring for which the probability that a random walk starting from a red vertex stays within the red class for at least $2$ steps is $x$, and the corresponding probability for blue is $y$? Our most general result is that for any $d$-regular graph, these pairs lie within the convex hull of the $2d$ points $\left\{\left(\frac{l}{d},\frac{l^2}{d^2}\right),\left(\frac{l^2}{d^2},\frac{l}{d}\right) :0\leq l\leq d\right\}$. Our main focus is the torus for which we prove both sharper bounds and existence results via constructions. In particular, for the $2$-dimensional torus, we show that asymptotically, the region in which these pairs of probabilities can lie is exactly the convex hull of: \[ \left\{\left(0,0\right),\left(\frac{1}{2},\frac{1}{4}\right),\left(\frac{3}{4},\frac{9}{16}\right),\left(\frac{1}{4},\frac{1}{2}\right),\left(\frac{9}{16},\frac{3}{4}\right),\left(1,1\right)\right\} \]

##### 4.Relative Fractional Independence Number and Its Applications

**Authors:**Sharareh Alipour, Amin Gohari

**Abstract:** We define the relative fractional independence number of two graphs, $G$ and $H$, as $$\alpha^*(G|H)=\max_{W}\frac{\alpha(G\boxtimes W)}{\alpha(H\boxtimes W)},$$ where the maximum is taken over all graphs $W$, $G\boxtimes W$ is the strong product of $G$ and $W$, and $\alpha$ denotes the independence number. We give a non-trivial linear program to compute $\alpha^*(G|H)$ and discuss some of its properties. We show that $$\alpha^*(G|H)\geq \frac{X(G)}{X(H)},$$ where $X(G)$ can be the independence number, the zero-error Shannon capacity, the fractional independence number, the Lov'{a}sz number, or the Schrijver's or Szegedy's variants of the Lov'{a}sz number of a graph $G$. This inequality is the first explicit non-trivial upper bound on the ratio of the invariants of two arbitrary graphs, as mentioned earlier, which can also be used to obtain upper or lower bounds for these invariants. As explicit applications, we present new upper bounds for the ratio of the zero-error Shannon capacity of two Cayley graphs and compute the Haemers number of certain Johnson graphs. Moreover, we show that the relative fractional independence number can be used to present a stronger version of the well-known No-Homomorphism Lemma. The No-Homomorphism Lemma is widely used to show the non-existence of a homomorphism between two graphs and is also used to give an upper bound on the independence number of a graph. Our extension of the No-Homomorphism Lemma is computationally more accessible than its original version.

##### 5.Smoothed Analysis of the Komlós Conjecture: Rademacher Noise

**Authors:**Elad Aigner-Horev, Dan Hefetz, Michael Trushkin

**Abstract:** The {\em discrepancy} of a matrix $M \in \mathbb{R}^{d \times n}$ is given by $\mathrm{DISC}(M) := \min_{\boldsymbol{x} \in \{-1,1\}^n} \|M\boldsymbol{x}\|_\infty$. An outstanding conjecture, attributed to Koml\'os, stipulates that $\mathrm{DISC}(M) = O(1)$, whenever $M$ is a Koml\'os matrix, that is, whenever every column of $M$ lies within the unit sphere. Our main result asserts that $\mathrm{DISC}(M + R/\sqrt{d}) \leq 1 + O(d^{-1/2})$ holds asymptotically almost surely, whenever $M \in \mathbb{R}^{d \times n}$ is Koml\'os, $R \in \mathbb{R}^{d \times n}$ is a Rademacher random matrix, $d = \omega(1)$, and $n = \tilde \omega(d^{5/4})$. We conjecture that $n = \omega(d \log d)$ suffices for the same assertion to hold. The factor $d^{-1/2}$ normalising $R$ is essentially best possible.

##### 1.Multi-parameter Szemerédi-Trotter-type theorems and applications in finite fields

**Authors:**Hung Le, Steven Senger, Minh-Quan Vo

**Abstract:** We prove some novel multi-parameter point-line incidence estimates in vector spaces over finite fields. While these could be seen as special cases of higher-dimensional incidence results, they outperform their more general counterparts in those contexts. We go on to present a number of applications to illustrate their use in combinatorial problems from geometry and number theory.

##### 2.Independent domination versus packing in subcubic graphs

**Authors:**Eun-Kyung Cho, Minki Kim

**Abstract:** In 2011, Henning, L\"{o}wenstein, and Rautenbach observed that the domination number of a graph is bounded from above by the product of the packing number and the maximum degree of the graph. We prove a stronger statement in subcubic graphs: the independent domination number is bounded from above by three times the packing number.

##### 3.Total mutual-visibility in Hamming graphs

**Authors:**Csilla Bujtá, Sandi Klavžar, Jing Tian

**Abstract:** If $G$ is a graph and $X\subseteq V(G)$, then $X$ is a total mutual-visibility set if every pair of vertices $x$ and $y$ of $G$ admits a shortest $x,y$-path $P$ with $V(P) \cap X \subseteq \{x,y\}$. The cardinality of a largest total mutual-visibility set of $G$ is the total mutual-visibility number $\mu_{\rm t}(G)$ of $G$. In this paper the total mutual-visibility number is studied on Hamming graphs, that is, Cartesian products of complete graphs. Different equivalent formulations for the problem are derived. The values $\mu_{\rm t}(K_{n_1}\,\square\, K_{n_2}\,\square\, K_{n_3})$ are determined. It is proved that $\mu_{\rm t}(K_{n_1} \,\square\, \cdots \,\square\, K_{n_r}) = O(N^{r-2})$, where $N = n_1+\cdots + n_r$, and that $\mu_{\rm t}(K_s^{\,\square\,, r}) = \Theta(s^{r-2})$ for every $r\ge 3$, where $K_s^{\,\square\,, r}$ denotes the Cartesian product of $r$ copies of $K_s$. The main theorems are also reformulated as Tur\'an-type results on hypergraphs.

##### 4.An Inversion Statistic on the Hyperoctahedral Group

**Authors:**Hasan Arslan, Alnour Altoum, Hilal Karakus Arslan

**Abstract:** In this paper, we introduce an inversion statistic on the hyperoctahedral group $B_n$ by using an decomposition of a positive root system of this reflection group. Then we prove some combinatorial properties for the inversion statistic. We establish an enumeration system on the group $B_n$ and give an efficient method to uniquely derive any group element known its enumeration order with the help of the inversion table. In addition, we prove that the \textit{flag-major index} is equi-distributed with this inversion statistic on $B_n$.

##### 5.Polytope Extensions with Linear Diameters

**Authors:**Volker Kaibel, Kirill Kukharenko

**Abstract:** We describe constructions of extended formulations that establish a certain relaxed version of the Hirsch-conjecture and prove that if there is a pivot rule for the simplex algorithm for which one can bound the number of steps by the (monotone) diameter of the polyhedron of feasible solutions then the general linear programming problem can be solved in strongly polynomial time.

##### 6.Pull-Push Method: A new approach to Edge-Isoperimetric Problems

**Authors:**Sergei L. Bezrukov, Nikola Kuzmanovski, Jounglag Lim

**Abstract:** We prove a generalization of the Ahlswede-Cai local-global principle. A new technique to handle edge-isoperimetric problems is introduced which we call the pull-push method. Our main result includes all previously published results in this area as special cases with the only exception of the edge-isoperimetric problem for grids. With this we partially answer a question of Harper on local-global principles. We also describe a strategy for further generalization of our results so that the case of grids would be covered, which would completely settle Harper's question.

##### 7.Reflect-Push Methods Part I: Two Dimensional Techniques

**Authors:**Nikola Kuzmanovski, Jamie Radcliffe

**Abstract:** We determine all maximum weight downsets in the product of two chains, where the weight function is a strictly increasing function of the rank. Many discrete isoperimetric problems can be reduced to the maximum weight downset problem. Our results generalize Lindsay's edge-isoperimetric theorem in two dimensions in several directions. They also imply and strengthen (in several directions) a result of Ahlswede and Katona concerning graphs with maximal number of adjacent pairs of edges. We find all optimal shifted graphs in the Ahlswede-Katona problem. Furthermore, the results of Ahlswede-Katona are extended to posets with a rank increasing and rank constant weight function. Our results also strengthen a special case of a recent result by Keough and Radcliffe concerning graphs with the fewest matchings. All of these results are achieved by applications of a key lemma that we call the reflect-push method. This method is geometric and combinatorial. Most of the literature on edge-isoperimetric inequalities focuses on finding a solution, and there are no general methods for finding all possible solutions. Our results give a general approach for finding all compressed solutions for the above edge-isoperimetric problems. By using the Ahlswede-Cai local-global principle, one can conclude that lexicographic solutions are optimal for many cases of higher dimensional isoperimetric problems. With this and our two dimensional results we can prove Lindsay's edge-isoperimetric inequality in any dimension. Furthermore, our results show that lexicographic solutions are the unique solutions for which compression techniques can be applied in this general setting.

##### 8.New infinite family of regular edge-isoperimetric graphs

**Authors:**Sergei L. Bezrukov, Pavle Bulatovic, Nikola Kuzmanovski

**Abstract:** We introduce a new infinite family of regular graphs admitting nested solutions in the edge-isoperimetric problem for all their Cartesian powers. The obtained results include as special cases most of previously known results in this area.

##### 9.Improved bounds for the Erdős-Rogers $(s,s+2)$-problem

**Authors:**Oliver Janzer, Benny Sudakov

**Abstract:** For $2\leq s<t$, the Erd\H{o}s-Rogers function $f_{s,t}(n)$ measures how large a $K_s$-free induced subgraph there must be in a $K_t$-free graph on $n$ vertices. There has been a tremendeus amount of work towards estimating this function, but until very recently only the case $t=s+1$ was well understood. A recent breakthrough of Mattheus and Verstra\"ete on the Ramsey number $r(4,k)$ states that $f_{2,4}(n)\leq n^{1/3+o(1)}$, which matches the known lower bound up to the $o(1)$ term. In this paper we build on their approach and generalize this result by proving that $f_{s,s+2}(n)\leq n^{\frac{2s-3}{4s-5}+o(1)}$ holds for every $s\geq 2$. This comes close to the best known lower bound, improves a substantial body of work and is the best that any construction of similar kind can give.

##### 10.Strictly $k$-colorable graphs

**Authors:**Evan Leonard

**Abstract:** Xuding Zhu introduced a refined scale of choosability in 2020 and observed that the four color theorem is tight on this scale. We formalize and explore this idea of tightness in what we call strictly colorable graphs. We then characterize all strictly colorable complete multipartite graphs.

##### 1.Exact generalized Turán number for $K_3$ versus suspension of $P_4$

**Authors:**Sayan Mukherjee

**Abstract:** Let $P_4$ denote the path graph on $4$ vertices. The suspension of $P_4$, denoted by $\widehat P_4$, is the graph obtained via adding an extra vertex and joining it to all four vertices of $P_4$. In this note, we demonstrate that for $n\ge 8$, the maximum number of triangles in any $n$-vertex graph not containing $\widehat P_4$ is $\left\lfloor n^2/8\right\rfloor$. Our method uses simple induction along with computer programming to prove a base case of the induction hypothesis.

##### 2.The Topological Quandles up to Four Elements

**Authors:**Mohamed Ayadi
LMBP

**Abstract:** The finite topological quandles can be represented as $n\times n$ matrices, recently defined by S. Nelson and C. Wong. In this paper, we first study the finite topological quandles and we show how to use these matrices to distinguish all isomorphism classes of finite topological quandles for a given cardinality $n$. As an application, we classify finite topological quandles with up to 4 elements.

##### 3.Bounding the chromatic number of tournaments by arc neighborhoods

**Authors:**Felix Klingelhoefer, Alantha Newman

**Abstract:** The chromatic number of a directed graph is the minimum number of induced acyclic subdigraphs that cover its vertex set, and accordingly, the chromatic number of a tournament is the minimum number of transitive subtournaments that cover its vertex set. The neighborhood of an arc $uv$ in a tournament $T$ is the set of vertices that form a directed triangle with arc $uv$. We show that if the neighborhood of every arc in a tournament has bounded chromatic number, then the whole tournament has bounded chromatic number. We show that this holds more generally for oriented graphs with bounded independence number, which we use to prove the equivalence of a conjecture of El-Zahar and Erd\H{o}s and a recent conjecture of Nguyen, Scott and Seymour relating the structure of graphs and tournaments with high chromatic number.

##### 4.Combinatorial Nullstellensatz and Turán numbers of complete $r$-partite $r$-uniform hypergraphs

**Authors:**Alexey Gordeev

**Abstract:** In this note we describe how Laso\'n's generalization of Alon's Combinatorial Nullstellensatz gives a framework for constructing lower bounds on the Tur\'an number $\operatorname{ex}(n, K^{(r)}_{s_1,\dots,s_r})$ of the complete $r$-partite $r$-uniform hypergraph $K^{(r)}_{s_1,\dots,s_r}$. To illustrate the potential of this method, we give a short and simple explicit construction for the Erd\H{o}s box problem, showing that $\operatorname{ex}(n, K^{(r)}_{2,\dots,2}) = \Omega(n^{r - 1/r})$, which asymptotically matches best known bounds when $r \leq 4$.

##### 5.Globally linked pairs of vertices in generic frameworks

**Authors:**Tibor Jordán, Soma Villányi

**Abstract:** A $d$-dimensional framework is a pair $(G,p)$, where $G=(V,E)$ is a graph and $p$ is a map from $V$ to $\mathbb{R}^d$. The length of an edge $xy\in E$ in $(G,p)$ is the distance between $p(x)$ and $p(y)$. A vertex pair $\{u,v\}$ of $G$ is said to be globally linked in $(G,p)$ if the distance between $p(u)$ and $p(v)$ is equal to the distance between $q(u)$ and $q(v)$ for every $d$-dimensional framework $(G,q)$ in which the corresponding edge lengths are the same as in $(G,p)$. We call $(G,p)$ globally rigid in $\mathbb{R}^d$ when each vertex pair of $G$ is globally linked in $(G,p)$. A pair $\{u,v\}$ of vertices of $G$ is said to be weakly globally linked in $G$ in $\mathbb{R}^d$ if there exists a generic framework $(G,p)$ in which $\{u,v\}$ is globally linked. In this paper we first give a sufficient condition for the weak global linkedness of a vertex pair of a $(d+1)$-connected graph $G$ in $\mathbb{R}^d$ and then show that for $d=2$ it is also necessary. We use this result to obtain a complete characterization of weakly globally linked pairs in graphs in $\mathbb{R}^2$, which gives rise to an algorithm for testing weak global linkedness in the plane in $O(|V|^2)$ time. Our methods lead to a new short proof for the characterization of globally rigid graphs in $\mathbb{R}^2$, and further results on weakly globally linked pairs and globally rigid graphs in the plane and in higher dimensions.

##### 6.The Pairing-Hamiltonian property in graph prisms

**Authors:**Marién Abreu, Giuseppe Mazzuoccolo, Federico Romaniello, Jean Paul Zerafa

**Abstract:** Let $G$ be a graph of even order, and consider $K_G$ as the complete graph on the same vertex set as $G$. A perfect matching of $K_G$ is called a pairing of $G$. If for every pairing $M$ of $G$ it is possible to find a perfect matching $N$ of $G$ such that $M \cup N$ is a Hamiltonian cycle of $K_G$, then $G$ is said to have the Pairing-Hamiltonian property, or PH-property, for short. In 2007, Fink [J. Combin. Theory Ser. B, 97] proved that for every $d\geq 2$, the $d$-dimensional hypercube $\mathcal{Q}_d$ has the PH-property, thus proving a conjecture posed by Kreweras in 1996. In this paper we extend Fink's result by proving that given a graph $G$ having the PH-property, the prism graph $\mathcal{P}(G)$ of $G$ has the PH-property as well. Moreover, if $G$ is a connected graph, we show that there exists a positive integer $k_0$ such that the $k^{\textrm{th}}$-prism of a graph $\mathcal{P}^k(G)$ has the PH-property for all $k \ge k_0$.

##### 7.Extremal numbers and Sidorenko's conjecture

**Authors:**David Conlon, Joonkyung Lee, Alexander Sidorenko

**Abstract:** Sidorenko's conjecture states that, for all bipartite graphs $H$, quasirandom graphs contain asymptotically the minimum number of copies of $H$ taken over all graphs with the same order and edge density. While still open for graphs, the analogous statement is known to be false for hypergraphs. We show that there is some advantage in this, in that if Sidorenko's conjecture does not hold for a particular $r$-partite $r$-uniform hypergraph $H$, then it is possible to improve the standard lower bound, coming from the probabilistic deletion method, for its extremal number $\mathrm{ex}(n,H)$, the maximum number of edges in an $n$-vertex $H$-free $r$-uniform hypergraph. With this application in mind, we find a range of new counterexamples to the conjecture for hypergraphs, including all linear hypergraphs containing a loose triangle and all $3$-partite $3$-uniform tight cycles.

##### 8.Boolean prism permutations in the Bruhat order

**Authors:**Bridget Eileen Tenner

**Abstract:** The boolean elements of a Coxeter group have been characterized and shown to possess many interesting properties and applications. Here we introduce "boolean prisms," a generalization of those elements, characterizing them equivalently in terms of their reduced words and in terms of pattern containment. As part of this work, we introduce the notion of "calibration" to permutation patterns.

##### 9.Winding number and circular 4-coloring of signed graphs

**Authors:**Anna Gujgiczer, Reza Naserasr, Rohini S, S Taruni

**Abstract:** Concerning the recent notion of circular chromatic number of signed graphs, for each given integer $k$ we introduce two signed bipartite graphs, each on $2k^2-k+1$ vertices, having shortest negative cycle of length $2k$, and the circular chromatic number 4. Each of the construction can be viewed as a bipartite analogue of the generalized Mycielski graphs on odd cycles, $M_{\ell}(C_{2k+1})$. In the course of proving our result, we also obtain a simple proof of the fact that $M_{\ell}(C_{2k+1})$ and some similar quadrangulations of the projective plane have circular chromatic number 4. These proofs have the advantage that they illuminate, in an elementary manner, the strong relation between algebraic topology and graph coloring problems.

##### 10.On tricyclic graphs with maximum edge Mostar index

**Authors:**Fazal Hayat, Shou-jun Xu, Bo Zhou

**Abstract:** For a given connected graph $G$, the edge Mostar index $Mo_e(G)$ is defined as $Mo_e(G)=\sum_{e=uv \in E(G)}|m_u(e|G) - m_v(e|G)|$, where $m_u(e|G)$ and $m_v(e|G)$ are respectively, the number of edges of $G$ lying closer to vertex $u$ than to vertex $v$ and the number of edges of $G$ lying closer to vertex $v$ than to vertex $u$. In this paper, we determine the sharp upper bound for the edge Mostar index on tricyclic graphs with a fixed number of edges, and the graphs that attain the bound are completely characterized.

##### 11.On the image of graph distance matrices

**Authors:**William Dudarov, Noah Feinberg, Raymond Guo, Ansel Goh, Andrea Ottolini, Alicia Stepin, Raghavenda Tripathi, Joia Zhang

**Abstract:** Let $G=(V,E)$ be a finite, simple, connected, combinatorial graph on $n$ vertices and let $D \in \mathbb{R}^{n \times n}$ be its graph distance matrix $D_{ij} = d(v_i, v_j)$. Steinerberger (J. Graph Theory, 2023) empirically observed that the linear system of equations $Dx =\mathbf{1}$, where $\mathbf{1} = (1,1,\dots, 1)^{T}$, very frequently has a solution (even in cases where $D$ is not invertible). The smallest nontrivial example of a graph where the linear system is not solvable are two graphs on 7 vertices. We prove that, in fact, counterexamples exists for all $n\geq 7$. The construction is somewhat delicate and further suggests that such examples are perhaps rare. We also prove that for Erd\H{o}s-R\'enyi random graphs the graph distance matrix $D$ is invertible with high probability. We conclude with some structural results on the Perron-Frobenius eigenvector for a distance matrix.

##### 1.Chip-firing on graphs of groups

**Authors:**Margaret Meyer, Dmitry Zakharov

**Abstract:** We define the Laplacian matrix and the Jacobian group of a finite graph of groups. We prove analogues of the matrix tree theorem and the class number formula for the order of the Jacobian of a graph of groups. Given a group $G$ acting on a graph $X$, we define natural pushforward and pullback maps between the Jacobian groups of $X$ and the quotient graph of groups $X/\!/G$. For the case $G=\mathbb{Z}/2\mathbb{Z}$, we also prove a combinatorial formula for the order of the kernel of the pushforward map.

##### 2.Realizing the $s$-permutahedron via flow polytopes

**Authors:**Rafael S. González D'León, Alejandro H. Morales, Eva Philippe, Daniel Tamayo Jiménez, Martha Yip

**Abstract:** Ceballos and Pons introduced the $s$-weak order on $s$-decreasing trees, for any weak composition $s$. They proved that it has a lattice structure and further conjectured that it can be realized as the $1$-skeleton of a polyhedral subdivision of a polytope. We answer their conjecture in the case where $s$ is a strict composition by providing three geometric realizations of the $s$-permutahedron. The first one is the dual graph of a triangulation of a flow polytope of high dimension. The second one, obtained using the Cayley trick, is the dual graph of a fine mixed subdivision of a sum of hypercubes that has the conjectured dimension. The third one, obtained using tropical geometry, is the $1$-skeleton of a polyhedral complex for which we can provide explicit coordinates of the vertices and whose support is a permutahedron as conjectured.

##### 3.On the uniqueness of collections of pennies and marbles

**Authors:**Sean Dewar, Georg Grasegger, Kaie Kubjas, Fatemeh Mohammadi, Anthony Nixon

**Abstract:** In this note we study the uniqueness problem for collections of pennies and marbles. More generally, consider a collection of unit $d$-spheres that may touch but not overlap. Given the existence of such a collection, one may analyse the contact graph of the collection. In particular we consider the uniqueness of the collection arising from the contact graph. Using the language of graph rigidity theory, we prove a precise characterisation of uniqueness (global rigidity) in dimensions 2 and 3 when the contact graph is additionally chordal. We then illustrate a wide range of examples in these cases. That is, we illustrate collections of marbles and pennies that can be perturbed continuously (flexible), are locally unique (rigid) and are unique (globally rigid). We also contrast these examples with the usual generic setting of graph rigidity.

##### 4.A Note On The Cross-Sperner Families

**Authors:**Junyao Pan

**Abstract:** Let $\mathcal{F}$ and $\mathcal{G}$ be two families of $[n]$, where $[n]=\{1,2,...,n\}$. If $A\not\subset B$ and $B\not\subset A$ hold for all $A\in\mathcal{F}$ and $B\in\mathcal{G}$, P. Frankl and Jian Wang called that $\mathcal{F}$ and $\mathcal{G}$ are cross-Sperner. Set $\mathcal{I}(\mathcal{F},\mathcal{G})=\{A\cap B:A\in\mathcal{F},B\in\mathcal{G}\}$. Define $m(n)={\rm{max}}\{|\mathcal{I}(\mathcal{F},\mathcal{G})|:\mathcal{F},\mathcal{G}\subset2^{[n]}~{\rm{are~cross}}$-${\rm{sperner}}\}$. In this note, we prove that $2^n-2^{\lfloor\frac{n}{2}\rfloor}-2^{\lceil\frac{n}{2}\rceil}+1$ for $n>1$. This not only solves an open problem proposed by P. Frankl and Jian Wang but also completes the extremal problem of cross-sperner families.

##### 5.On $m$-ovoids of $Q^+(7,q)$ with $q$ odd

**Authors:**Sam Adriaensen, Jan De Beule, Giovanni Giuseppe Grimaldi, Jonathan Mannaert

**Abstract:** In this paper, we provide a construction of $(q+1)$-ovoids of the hyperbolic quadric $Q^+(7,q)$, $q$ an odd prime power, by glueing $(q+1)/2$-ovoids of the elliptic quadric $Q^-(5,q)$. This is possible by controlling some intersection properties of (putative) $m$-ovoids of elliptic quadrics. It yields eventually $(q+1)$-ovoids of $Q^+(7,q)$ not coming from a $1$-system. Secondly, we also construct $m$-ovoids for $m \in \{ 2,4,6,8,10\}$ in $Q^+(7,3)$. Therefore we first investigate how to construct spreads of $\pg(3,q)$ that have as many secants to an elliptic quadric as possible.

##### 6.Off-Diagonal Commonality of Graphs via Entropy

**Authors:**Natalie Behague, Natasha Morrison, Jonathan A. Noel

**Abstract:** A graph $H$ is common if the limit as $n\to\infty$ of the minimum density of monochromatic labelled copies of $H$ in an edge colouring of $K_n$ with red and blue is attained by a sequence of quasirandom colourings. We apply an information-theoretic approach to show that certain graphs obtained from odd cycles and paths via gluing operations are common. In fact, for every pair $(H_1,H_2)$ of such graphs, there exists $p\in(0,1)$ such that an appropriate linear combination of red copies of $H_1$ and blue copies of $H_2$ is minimized by a quasirandom colouring in which $p\binom{n}{2}$ edges are red; such a pair $(H_1,H_2)$ is said to be $(p,1-p)$-common. Our approach exploits a strengthening of the common graph property for odd cycles that was recently proved using Schur convexity. We also exhibit a $(p,1-p)$-common pair $(H_1,H_2)$ such that $H_2$ is uncommon.

##### 7.On the eccentric graph of trees

**Authors:**Sezer Sorgun, Esma Elyemani

**Abstract:** We consider the eccentric graph of a graph $G$, denoted by $ecc(G)$, which has the same vertex set as $G$, and two vertices in the eccentric graph are adjacent iff their distance in $G$ is equal to the eccentricity of one of them. In this paper, we present a fundamental requirement for the isomorphism between $ecc(G)$ and the complement of $G$, and show that the previous necessary condition given in the literature is inadequate. Also we obtain that diameter of $ecc(T)$ is at most $3$ for any tree and get some characterizations of the eccentric graph of trees.

##### 1.On the $\operatorname{rix}$ statistic and valley-hopping

**Authors:**Nadia Lafrenière, Yan Zhuang

**Abstract:** This paper studies the relationship between the modified Foata$\unicode{x2013}$Strehl action (a.k.a. valley-hopping)$\unicode{x2014}$a group action on permutations used to demonstrate the $\gamma$-positivity of the Eulerian polynomials$\unicode{x2014}$and the number of rixed points $\operatorname{rix}$$\unicode{x2014}$a recursively-defined permutation statistic introduced by Lin in the context of an equidistribution problem. We give a linear-time iterative algorithm for computing the set of rixed points, and prove that the $\operatorname{rix}$ statistic is homomesic under valley-hopping. We also demonstrate that a bijection $\Phi$ introduced by Lin and Zeng in the study of the $\operatorname{rix}$ statistic sends orbits of the valley-hopping action to orbits of a cyclic version of valley-hopping, which implies that the number of fixed points $\operatorname{fix}$ is homomesic under cyclic valley-hopping.

##### 2.A short proof of Seymour's 6-flow theorem

**Authors:**Matt DeVos, Kathryn Nurse

**Abstract:** We give a compact variation of Seymour's proof that every $2$-edge-connected graph has a nowhere-zero $\mathbb{Z}_2 \times \mathbb{Z}_3$-flow.

##### 3.The grid-minor theorem revisited

**Authors:**Vida Dujmović, Robert Hickingbotham, Jędrzej Hodor, Gweanël Joret, Hoang La, Piotr Micek, Pat Morin, Clément Rambaud, David R. Wood

**Abstract:** We prove that for every planar graph $X$ of treedepth $h$, there exists a positive integer $c$ such that for every $X$-minor-free graph $G$, there exists a graph $H$ of treewidth at most $f(h)$ such that $G$ is isomorphic to a subgraph of $H\boxtimes K_c$. This is a qualitative strengthening of the Grid-Minor Theorem of Robertson and Seymour (JCTB 1986), and treedepth is the optimal parameter in such a result. As an example application, we use this result to improve the upper bound for weak coloring numbers of graphs excluding a fixed graph as a minor.

##### 4.Restricting Dyck Paths and 312-avoiding Permutations

**Authors:**Elena Barcucci, Antonio Bernini, Stefano Bilotta, Renzo Pinzani

**Abstract:** Dyck paths having height at most $h$ and without valleys at height $h-1$ are combinatorially interpreted by means of 312-avoding permutations with some restrictions on their \emph{left-to-right maxima}. The results are obtained by analyzing a restriction of a well-known bijection between the sets of Dyck paths and 312-avoding permutations. We also provide a recursive formula enumerating these two structures using ECO method and the theory of production matrices. As a further result we obtain a family of combinatorial identities involving Catalan numbers.

##### 5.On cocliques in commutative Schurian association schemes of the symmetric group

**Authors:**Roghayeh Maleki, Andriaherimanana Sarobidy Razafimahatratra

**Abstract:** Given the symmetric group $G = \operatorname{Sym}(n)$ and a multiplicity-free subgroup $H\leq G$, the orbitals of the action of $G$ on $G/H$ by left multiplication induce a commutative association scheme. The irreducible constituents of the permutation character of $G$ acting on $G/H$ are indexed by partitions of $n$ and if $\lambda \vdash n$ is the second largest partition in dominance ordering among these, then the Young subgroup $\operatorname{Sym}(\lambda)$ admits two orbits in its action on $G/H$, which are $\mathcal{S}_\lambda$ and its complement. In their monograph [Erd\H{o}s-Ko-Rado theorems: Algebraic Approaches. {\it Cambridge University Press}, 2016] (Problem~16.13.1), Godsil and Meagher asked whether $\mathcal{S}_\lambda$ is a coclique of a graph in the commutative association scheme arising from the action of $G$ on $G/H$. If such a graph exists, then they also asked whether its smallest eigenvalue is afforded by the $\lambda$-module. In this paper, we initiate the study of this question by taking $\lambda = [n-1,1]$. We show that the answer to this question is affirmative for the pair of groups $\left(G,H\right)$, where $G = \operatorname{Sym}(2k+1)$ and $H = \operatorname{Sym}(2) \wr \operatorname{Sym}(k)$, or $G = \operatorname{Sym}(n)$ and $H$ is one of $\operatorname{Alt}(k) \times \operatorname{Sym}(n-k),\ \operatorname{Alt}(k) \times \operatorname{Alt}(n-k)$, or $\left(\operatorname{Alt}(k)\times \operatorname{Alt}(n-k)\right) \cap \operatorname{Alt}(n)$. For the pair $(G,H) = \left(\operatorname{Sym}(2k),\operatorname{Sym}(k)\wr \operatorname{Sym}(2)\right)$, we also prove that the answer to this question of Godsil and Meagher is negative.

##### 6.Hypergraphs with arbitrarily small codegree Turán density

**Authors:**Simón Piga, Bjarne Schülke

**Abstract:** Let $k\geq 3$. Given a $k$-uniform hypergraph $H$, the minimum codegree $\delta(H)$ is the largest $d\in\mathbb{N}$ such that every $(k-1)$-set of $V(H)$ is contained in at least $d$ edges. Given a $k$-uniform hypergraph $F$, the codegree Tur\'an density $\gamma(F)$ of $F$ is the smallest $\gamma \in [0,1]$ such that every $k$-uniform hypergraph on $n$ vertices with $\delta(H)\geq (\gamma + o(1))n$ contains a copy of $F$. Similarly as other variants of the hypergraph Tur\'an problem, determining the codegree Tur\'an density of a hypergraph is in general notoriously difficult and only few results are known. In this work, we show that for every $\varepsilon>0$, there is a $k$-uniform hypergraph $F$ with $0<\gamma(F)<\varepsilon$. This is in contrast to the classical Tur\'an density, which cannot take any value in the interval $(0,k!/k^k)$ due to a fundamental result by Erd\H{o}s.

##### 7.Laplacian Spectra of Semigraphs

**Authors:**Pralhad M. Shinde

**Abstract:** Consider a semigraph $G=(V,\,E)$; in this paper, we study the eigenvalues of the Laplacian matrix of $G$. We show that the Laplacian of $G$ is positive semi-definite, and $G$ is connected if and only if $\lambda_2 >0.$ Along the similar lines of graph theory bounds on the largest eigenvalue, we obtain upper and lower bounds on the largest Laplacian eigenvalue of G and enumerate the Laplacian eigenvalues of some special semigraphs such as star semigraph, rooted 3-uniform semigraph tree.

##### 8.Lower (total) mutual visibility in graphs

**Authors:**Boštjan Brešar, Ismael G. Yero

**Abstract:** Given a graph $G$, a set $X$ of vertices in $G$ satisfying that between every two vertices in $X$ (respectively, in $G$) there is a shortest path whose internal vertices are not in $X$ is a mutual-visibility (respectively, total mutual-visibility) set in $G$. The cardinality of a largest (total) mutual-visibility set in $G$ is known under the name (total) mutual-visibility number, and has been studied in several recent works. In this paper, we propose two lower variants of the mentioned concepts, defined as the smallest possible cardinality among all maximal (total) mutual-visibility sets in $G$, and denote them by $\mu^{-}(G)$ and $\mu_t^{-}(G)$, respectively. While the total mutual-visibility number is never larger than the mutual-visibility number in a graph $G$, we prove that both differences $\mu^{-}(G)-\mu_t^{-}(G)$ and $\mu_t^{-}(G)-\mu^{-}(G)$ can be arbitrarily large. We characterize graphs $G$ with some small values of $\mu^{-}(G)$ and $\mu_t^{-}(G)$, and prove a useful tool called Neighborhood Lemma, which enables us to find upper bounds on the lower mutual-visibility number in several classes of graphs. We compare the lower mutual-visibility number with the lower general position number, and find a close relationship with Bollob\'{a}s-Wessel theorem when this number is considered in Cartesian products of complete graphs. Finally, we also prove the NP-completeness of the decision problem related to $\mu_t^{-}(G)$.

##### 9.Tree expansions of some Lie idempotents}

**Authors:**Frédéric Menous, Jean-Christophe Novelli, Jean-Yves Thibon

**Abstract:** We prove that the Catalan Lie idempotent $D_n(a,b)$, introduced in [Menous {\it et al.}, Adv. Appl. Math. 51 (2013), 177] can be refined by introducing $n$ independent parameters $a_0,\ldots,a_{n-1}$ and that the coefficient of each monomial is itself a Lie idempotent in the descent algebra. These new idempotents are multiplicity-free sums of subsets of the Poincar\'e-Birkhoff-Witt basis of the Lie module. These results are obtained by embedding noncommutative symmetric functions into the dual noncommutative Connes-Kreimer algebra, which also allows us to interpret, and rederive in a simpler way, Chapoton's results on a two-parameter tree expanded series.

##### 10.Kemperman's inequality and Freiman's lemma via few translates

**Authors:**Yifan Jing, Akshat Mudgal

**Abstract:** Let $G$ be a connected compact group equipped with the normalised Haar measure $\mu$. Our first result shows that given $\alpha, \beta>0$, there is a constant $c = c(\alpha,\beta)>0$ such that for any compact sets $A,B\subseteq G$ with $ \alpha\mu(B)\geq\mu(A)\geq \mu(B) $ and $ \mu(A)+\mu(B)\leq 1-\beta$, there exist $b_1,\dots b_c\in B$ such that \[ \mu(A\cdot \{b_1,\dots,b_c\})\geq \mu(A)+\mu(B).\] A special case of this, that is, when $G=\mathbb{T}^d$, confirms a recent conjecture of Bollob\'as, Leader and Tiba. We also prove a quantitatively stronger version of such a result in the discrete setting of $\mathbb{R}^d$. Thus, given $d \in \mathbb{N}$, we show that there exists $c = c(d) >0$ such that for any finite, non-empty set $A \subseteq \mathbb{R}^d$ which is not contained in a translate of a hyperplane, one can find $a_1, \dots, a_c \in A$ satisfying \[ |A+ \{a_1, \dots, a_c\}| \geq (d+1)|A| - O_d(1). \] The main term here is optimal and recovers the bounds given by Freiman's lemma up to the $O_d(1)$ error term.

##### 11.Convex Hull Thrackles

**Authors:**Balázs Keszegh, Dániel Simon

**Abstract:** A \emph{thrackle} is a graph drawn in the plane so that every pair of its edges meet exactly once, either at a common end vertex or in a proper crossing. Conway's thrackle conjecture states that the number of edges is at most the number of vertices. It is known that this conjecture holds for linear thrackles, i.e., when the edges are drawn as straight line segments. We consider \emph{convex hull thrackles}, a recent generalization of linear thrackles from segments to convex hulls of subsets of points. We prove that if the points are in convex position then the number of convex hulls is at most the number of vertices, but in general there is a construction with one more convex hull. On the other hand, we prove that the number of convex hulls is always at most twice the number of vertices.

##### 12.A network flow approach to a common generalization of Clar and Fries numbers

**Authors:**Erika Bérczi-Kovács, András Frank

**Abstract:** Clar number and Fries number are two thoroughly investigated parameters of plane graphs emerging from mathematical chemistry to measure stability of organic molecules. We consider first a common generalization of these two concepts for bipartite plane graphs, and then extend it to a framework on general (not necessarily planar) directed graphs. The corresponding optimization problem can be transformed into a maximum weight feasible tension problem which is the linear programming dual of a minimum cost network flow (or circulation) problem. Therefore the approach gives rise to a min-max theorem and to a strongly polynomial algorithm that relies exclusively on standard network flow subroutines. In particular, we give the first network flow based algorithm for an optimal Fries structure and its variants.

##### 1.Oriented spanning trees and stationary distribution of digraphs

**Authors:**Jiang Zhou, Changjiang Bu

**Abstract:** By using biclique partitions of digraphs, this paper gives reduction formulas for the number of oriented spanning trees, stationary distribution vector and Kemeny's constant of digraphs. As applications, we give a method for enumerating spanning trees of undirected graphs by vertex degrees and biclique partitions. The biclique partition formula also extends the results of Knuth and Levine from line digraphs to general digraphs.

##### 2.Effective Bounds for the Partition Function Weighted by the Parity of the Crank and Applications

**Authors:**Janet J. W. Dong, Kathy Q. Ji

**Abstract:** Let $M_0(n)$ (resp. $M_1(n)$) denote the number of partitions of $n$ with even (reps. odd) crank. We establish the effective bound for the asymptotic formula for $M_0(n)-M_1(n)$ due to Choi, Kang and Lovejoy. By utilizing this formula with the explicit bound, we demonstrate that the cranks are asymptotically equidistributed modulo 2. We also use this formula to show that $(-1)^n(M_0(n)-M_1(n))>d$ for $d\geq 1$ and $n\geq \left\lceil\frac{24}{\pi^2}\left(\ln\left(\frac{7d}{2}\right)\right)^2+\frac{1}{24}\right\rceil$ which strengthens a result due to Andrews and Lewis. Moreover, we establish an upper bound and a lower bound for $M_0(n)$ and $M_1(n)$. The upper bound and the lower bound for $M_k(n)$ enable us to show that $M_k(n-1)+M_k(n+1)>2M_k(n)$ for $k=0$ or $1$ and $n\geq 39$. This result can be seen as the refinement of the classical result regarding the convexity of the partition function $p(n)$, which counts the number of partitions of $n$. We also show that $M_0(n)$ (resp. $M_1(n)$) is log-concave for $n\geq 94$ and satisfies the higher order Tur\'an inequalities for $n\geq 207$ with the aid of the upper bound and the lower bound for $M_0(n)$ and $M_1(n)$.

##### 1.Graphs with girth 9 and without longer odd holes are 3-colorable

**Authors:**Yan Wang, Rong Wu

**Abstract:** For a number $l\geq 2$, let ${\cal{G}}_l$ denote the family of graphs which have girth $2l+1$ and have no odd hole with length greater than $2l+1$. Wu, Xu and Xu conjectured that every graph in $\bigcup_{l\geq 2} {\cal{G}}_{l}$ is $3$-colorable. Chudnovsky et al., Wu et al., and Chen showed that every graph in ${\cal{G}}_2$, ${\cal{G}}_3$ and $\bigcup_{l\geq 5} {\cal{G}}_{l}$ is $3$-colorable respectively. In this paper, we prove that every graph in ${\cal{G}}_4$ is $3$-colorable. This confirms Wu, Xu and Xu's conjecture.

##### 2.On Hofstadter's G-sequence

**Authors:**Michel Dekking

**Abstract:** We characterize the entries of Hofstadter's G-sequence in terms of the lower and upper Wythoff sequences. This can be used as a tool to prove a longstanding conjecture on the averages of the swapped Wythoff sequences.

##### 1.The Number of Ribbon Tilings for Strips

**Authors:**Yinsong Chen, Vladislav Kargin

**Abstract:** First, we consider order-$n$ ribbon tilings of an $M$-by-$N$ rectangle $R_{M,N}$ where $M$ and $N$ are much larger than $n$. We prove the existence of the growth rate $\gamma_n$ of the number of tilings and show that $\gamma_n \leq (n-1) \ln 2$. Then, we study a rectangle $R_{M,N}$ with fixed width $M=n$, called a strip. We derive lower and upper bounds on the growth rate $\mu_n$ for strips as $ \ln n - 1 + o(1) \leq \mu_n \leq \ln n $. Besides, we construct a recursive system which enables us to enumerate the order-$n$ ribbon tilings of a strip for all $n \leq 8$ and calculate the corresponding generating functions.

##### 2.Induced subgraph density. II. Sparse and dense sets in cographs

**Authors:**Jacob Fox, Tung Nguyen, Alex Scott, Paul Seymour

**Abstract:** A well-known theorem of R\"odl says that for every graph $H$, and every $\epsilon>0$, there exists $\delta>0$ such that if $G$ does not contain an induced copy of $H$, then there exists $X\subseteq V(G)$ with $|X|\ge\delta|G|$ such that one of $G[X],\overline{G}[X]$ has edge-density at most $\epsilon$. But how does $\delta$ depend on $\epsilon$? Fox and Sudakov conjectured that the dependence can be taken to be polynomial: for all $H$ there exists $c>0$ such that for all $\epsilon\in(0,1/2)$, R\"odl's theorem holds with $\delta=\epsilon^c$. This conjecture implies the Erd\H{o}s-Hajnal conjecture, and until now it had not been verified for any non-trivial graphs $H$. Our first result shows that it is true when $H=P_4$ (indeed, we can take $\delta=\epsilon$, and insist that one of $G[X],\overline{G}[X]$ has maximum degree at most $\epsilon^2|G|$).) We will generalize this, and to do so, we need to work with an even stronger property. Let us say $H$ is {\em viral} if there exists $c>0$ such that for all $\epsilon\in(0,1/2)$, if $G$ contains at most $\epsilon^c|G|^{|H|}$ induced copies of $H$, then there exists $X\subseteq V(G)$ with $|X|\ge \epsilon^c|G|$ such that one of $G[X],\overline{G}[X]$ has density at most $\epsilon$. We will show that $P_4$ is viral, using a ``polynomial $P_4$-removal lemma'' of Alon and Fox. We will also show that viral graphs are closed under vertex-substitution, and so all graphs that can be obtained by substitution from copies of $P_4$ are viral. (In a subsequent paper, it will be shown that all graphs currently known to satisfy the Erd\H{o}s-Hajnal conjecture are in fact viral.) Finally, we give a different strengthening of R\"odl's theorem: we show that if $G$ does not contain an induced copy of $P_4$, then its vertices can be partitioned into at most $480\epsilon^{-4}$ subsets $X$ such that one of $G[X],\overline{G}[X]$ has density at most $\epsilon$.

##### 3.Hamilton transversals in tournaments

**Authors:**Debsoumya Chakraborti, Jaehoon Kim, Hyunwoo Lee, Jaehyeon Seo

**Abstract:** It is well-known that every tournament contains a Hamilton path, and every strongly connected tournament contains a Hamilton cycle. This paper establishes transversal generalizations of these classical results. For a collection $\mathbf{T}=\{T_1,\dots,T_m\}$ of not-necessarily distinct tournaments on a common vertex set $V$, an $m$-edge directed graph $\mathcal{D}$ with vertices in $V$ is called a $\mathbf{T}$-transversal if there exists a bijection $\phi\colon E(\mathcal{D})\to [m]$ such that $e\in E(T_{\phi(e)})$ for all $e\in E(\mathcal{D})$. We prove that for sufficiently large $m$ with $m=|V|-1$, there exists a $\mathbf{T}$-transversal Hamilton path. Moreover, if $m=|V|$ and at least $m-1$ of the tournaments $T_1,\ldots,T_m$ are assumed to be strongly connected, then there is a $\mathbf{T}$-transversal Hamilton cycle. In our proof, we utilize a novel way of partitioning tournaments which we dub $\mathbf{H}$-partition.

##### 4.On the second largest distance eigenvalues of graphs

**Authors:**Haiyan Guo, Bo Zhou

**Abstract:** The distance spectrum of a connected graph is the spectrum of its distance matrix. We characterize those bicyclic graphs and split graphs whose second largest distance eigenvalues belong to the interval $(-\infty,-\frac{1}{2})$. Any such graph is chordal.

##### 5.On the maximal Sombor index of quasi-tree graphs

**Authors:**Ruiting Zhang, Huiqing Liu, Yibo Li

**Abstract:** The Sombor index $SO(G)$ of a graph $G$ is the sum of the edge weights $\sqrt{d^2_G(u)+d^2_G(v)}$ of all edges $uv$ of $G$, where $d_G(u)$ denotes the degree of the vertex $u$ in $G$. A connected graph $G = (V ,E)$ is called a quasi-tree, if there exists $u\in V (G)$ such that $G-u$ is a tree. Denote $\mathscr{Q}(n,k)$=\{$G$: $G$ is a quasi-tree graph of order $n$ with $G-u$ being a tree and $d_G(u)=k$\}. In this paper, we determined the maximum, the second maximum and the third maximum Sombor indices of all quasi-tree graphs in $\mathscr{Q}(n,k)$, respectively. Moreover, we characterized their corresponding extremal graphs, respectively.

##### 6.Crystal isomorphisms and Mullineux involution II

**Authors:**Nicolas Jacon
LMR, Cédric Lecouvey
IDP

**Abstract:** We present a new combinatorial and conjectural algorithm for computing the Mullineux involution for the symmetric group and its Hecke algebra. This algorithm is built on a conjectural property of crystal isomorphisms which can be rephrased in a purely combinatorial way.

##### 7.Combinatorially rich sets in partial semigroups

**Authors:**Arpita Ghosh

**Abstract:** The notion of combinatorially rich set was first introduced by V. Bergelson and D. Glasscock. The goal of this paper is to study these subsets in the setting of adequate partial semigroups and describes its connection with the other concepts of large sets.

##### 8.Finding dense minors using average degree

**Authors:**Kevin Hendrey, Sergey Norin, Raphael Steiner, Jérémie Turcotte

**Abstract:** Motivated by Hadwiger's conjecture, we study the problem of finding the densest possible $t$-vertex minor in graphs of average degree at least $t-1$. We show that if $G$ has average degree at least $t-1$, it contains a minor on $t$ vertices with at least $(\sqrt{2}-1-o(1))\binom{t}{2}$ edges. We show that this cannot be improved beyond $\left(\frac{3}{4}+o(1)\right)\binom{t}{2}$. Finally, for $t\leq 6$ we exactly determine the number of edges we are guaranteed to find in the densest $t$-vertex minor in graphs of average degree at least $t-1$.

##### 9.Edge-coloring a graph $G$ so that every copy of a graph $H$ has an odd color class

**Authors:**Emily Heath, Shira Zerbib

**Abstract:** Recently, Alon introduced the notion of an $H$-code for a graph $H$: a collection of graphs on vertex set $[n]$ is an $H$-code if it contains no two members whose symmetric difference is isomorphic to $H$. Let $D_{H}(n)$ denote the maximum possible cardinality of an $H$-code, and let $d_{H}(n)=D_{H}(n)/2^{n \choose 2}$. Alon observed that a lower bound on $d_{H}(n)$ can be obtained by attaining an upper bound on the number of colors needed to edge-color $K_n$ so that every copy of $H$ has an odd color class. Motivated by this observation, we define $g(G,H)$ to be the minimum number of colors needed to edge-color a graph $G$ so that every copy of $H$ has an odd color class. We prove $g(K_n,K_5) \le n^{o(1)}$ and $g(K_{n,n}, C_4) \le \frac{t}{2(t-1)}n+o(n)$ for all odd integers $t\geq 5$. The first result shows $d_{K_5}(n) \ge \frac{1}{n^{o(1)}}$ and was obtained independently in arXiv:2306.14682.

##### 10.Another Proof of the Generalized Tutte--Berge Formula for $f$-Bounded Subgraphs

**Authors:**Zishen Qu, Douglas B. West

**Abstract:** Given a nonnegative integer weight $f(v)$ for each vertex $v$ in a multigraph $G$, an {\it $f$-bounded subgraph} of $G$ is a multigraph $H$ contained in $G$ such that $d_H(v)\le f(v)$ for all $v\in V(G)$. Using Tutte's $f$-Factor Theorem, we give a new proof of the min-max relation for the maximum size of an $f$-bounded subgraph of $G$. When $f(v)=1$ for all $v$, the formula reduces to the classical Tutte--Berge Formula for the maximum size of a matching.

##### 11.Sharp Hypercontractivity for Global Functions

**Authors:**Nathan Keller, Noam Lifshitz, Omri Marcus

**Abstract:** For a function $f$ on the hypercube $\{0,1\}^n$ with Fourier expansion $f=\sum_{S\subseteq[n]}\hat f(S)\chi_S$, the hypercontractive inequality for the noise operator allows bounding norms of $T_\rho f=\sum_S\rho^{|S|}\hat f(S)\chi_S$ in terms of norms of $f$. If $f$ is Boolean-valued, the level-$d$ inequality allows bounding the norm of $f^{=d}=\sum_{|S|=d}\hat f(S)\chi_S$ in terms of $E[f]$. These two inequalities play a central role in analysis of Boolean functions and its applications. While both inequalities hold in a sharp form when the hypercube is endowed with the uniform measure, they do not hold for more general discrete product spaces. Finding a `natural' generalization was a long-standing open problem. In [P. Keevash et al., Global hypercontractivity and its applications, J. Amer. Math. Soc., to appear], a hypercontractive inequality for this setting was presented, that holds for functions which are `global' -- namely, are not significantly affected by a restriction of a small set of coordinates. This hypercontractive inequality is not sharp, which precludes applications to the symmetric group $S_n$ and to other settings where sharpness of the bound is crucial. Also, no level-$d$ inequality for global functions over general discrete product spaces is known. We obtain sharp versions of the hypercontractive inequality and of the level-$d$ inequality for this setting. Our inequalities open the way for diverse applications to extremal set theory and to group theory. We demonstrate this by proving quantitative bounds on the size of intersecting families of sets and vectors under weak symmetry conditions and by describing numerous applications to the study of functions on $S_n$ -- including hypercontractivity and level-$d$ inequalities, character bounds, variants of Roth's theorem and of Bogolyubov's lemma, and diameter bounds, that were obtained using our techniques.

##### 12.On Restricted Intersections and the Sunflower Problem

**Authors:**Jeremy Chizewer

**Abstract:** A sunflower with $r$ petals is a collection of $r$ sets over a ground set $X$ such that every element in $X$ is in no set, every set, or exactly one set. Erd\H{o}s and Rado \cite{er} showed that a family of sets of size $n$ contains a sunflower if there are more than $n!(r-1)^n$ sets in the family. Alweiss et al. \cite{alwz} and subsequently, Rao \cite{rao} improved this bound to $(O(r \log(rn))^n$. We study the case where the pairwise intersections of the set family are restricted. In particular, we improve the best-known bound for set families when the size of the pairwise intersections of any two sets is in a set $L$. We also present a new bound for the special case when the set $L$ is the nonnegative integers less than or equal to $d$ using the techniques of Alweiss et al. \cite{alwz}.

##### 1.Computational Complexity in Algebraic Combinatorics

**Authors:**Greta Panova

**Abstract:** Algebraic Combinatorics originated in Algebra and Representation Theory, studying their discrete objects and integral quantities via combinatorial methods which have since developed independent and self-contained lives and brought us some beautiful formulas and combinatorial interpretations. The flagship hook-length formula counts the number of Standard Young Tableaux, which also gives the dimension of the irreducible Specht modules of the Symmetric group. The elegant Littlewood-Richardson rule gives the multiplicities of irreducible GL-modules in the tensor products of GL-modules. Such formulas and rules have inspired large areas of study and development beyond Algebra and Combinatorics, becoming applicable to Integrable Probability and Statistical Mechanics, and Computational Complexity Theory. We will see what lies beyond the reach of such nice product formulas and combinatorial interpretations and enter the realm of Computational Complexity Theory, that could formally explain the beauty we see and the difficulties we encounter in finding further formulas and ``combinatorial interpretations''. A 85-year-old such problem asks for a positive combinatorial formula for the Kronecker coefficients of the Symmetric group, another one pertains to the plethysm coefficients of the General Linear group. In the opposite direction, the study of Kronecker and plethysm coefficients leads to the disproof of the wishful approach of Geometric Complexity Theory (GCT) towards the resolution of the algebraic P vs NP Millennium problem, the VP vs VNP problem. In order to make GCT work and establish computational complexity lower bounds, we need to understand representation theoretic multiplicities in further detail, possibly asymptotically.

##### 2.A co-preLie structure from chronological loop erasure in graph walks

**Authors:**Loïc Foissy, Pierre-Louis Giscard, Cécile Mammez

**Abstract:** We show that the chronological removal of cycles from a walk on a graph, known as Lawler's loop-erasing procedure, generates a preLie co-algebra on the vector space spanned by the walks. In addition, we prove that the tensor and symmetric algebras of graph walks are graded Hopf algebras, provide their antipodes explicitly and recover the preLie co-algebra from a brace coalgebra on the tensor algebra of graph walks. Finally we exhibit sub-Hopf algebras associated to particular types of walks.

##### 3.Complete bipartite graphs without small rainbow stars

**Authors:**Weizhen Chen, Meng Ji, Yaping Mao, Meiqin Wei

**Abstract:** The $k$-edge-colored bipartite Gallai-Ramsey number $\operatorname{bgr}_k(G:H)$ is defined as the minimum integer $n$ such that $n^2\geq k$ and for every $N\geq n$, every edge-coloring (using all $k$ colors) of complete bipartite graph $K_{N,N}$ contains a rainbow copy of $G$ or a monochromatic copy of $H$. In this paper, we first study the structural theorem on the complete bipartite graph $K_{n,n}$ with no rainbow copy of $K_{1,3}$. Next, we utilize the results to prove the exact values of $\operatorname{bgr}_{k}(P_4: H)$, $\operatorname{bgr}_{k}(P_5: H)$, $\operatorname{bgr}_{k}(K_{1,3}: H)$, where $H$ is a various union of cycles and paths and stars.

##### 4.Borel Vizing's Theorem for Graphs of Subexponential Growth

**Authors:**Anton Bernshteyn, Abhishek Dhawan

**Abstract:** We show that every Borel graph $G$ of subexponential growth has a Borel proper edge-coloring with $\Delta(G) + 1$ colors. We deduce this from a stronger result, namely that an $n$-vertex (finite) graph $G$ of subexponential growth can be properly edge-colored using $\Delta(G) + 1$ colors by an $O(\log^\ast n)$-round deterministic distributed algorithm in the $\mathsf{LOCAL}$ model, where the implied constants in the $O(\cdot)$ notation are determined by a bound on the growth rate of $G$.

##### 5.Moment sequences, transformations, and Spidernet graphs

**Authors:**Paul Barry

**Abstract:** We use the link between Jacobi continued fractions and the generating functions of certain moment sequences to study some simple transformations on them. In particular, we define and study a transformation that is appropriate for the study of spidernet graphs and their moments, and the free Meixner law.

##### 6.The maximum number of odd cycles in a planar graph

**Authors:**Emily Heath, Ryan R. Martin, Chris Wells

**Abstract:** How many copies of a fixed odd cycle, $C_{2m+1}$, can a planar graph contain? We answer this question asymptotically for $m\in\{2,3,4\}$ and prove a bound which is tight up to a factor of $3/2$ for all other values of $m$. This extends the prior results of Cox--Martin and Lv et al. on the analogous question for even cycles. Our bounds result from a reduction to the following maximum likelihood question: which probability mass $\mu$ on the edges of some clique maximizes the probability that $m$ edges sampled independently from $\mu$ form either a cycle or a path?

##### 1.Spectral extremal graphs for edge blow-up of star forests

**Authors:**Jing Wang, Zhenyu Ni, Liying Kang, Yi-zheng Fan

**Abstract:** The edge blow-up of a graph $G$, denoted by $G^{p+1}$, is obtained by replacing each edge of $G$ with a clique of order $p+1$, where the new vertices of the cliques are all distinct. Yuan [J. Comb. Theory, Ser. B, 152 (2022) 379-398] determined the range of the Tur\'{a}n numbers for edge blow-up of all bipartite graphs and the exact Tur\'{a}n numbers for edge blow-up of all non-bipartite graphs. In this paper we prove that the graphs with the maximum spectral radius in an $n$-vertex graph without any copy of edge blow-up of star forests are the extremal graphs for edge blow-up of star forests when $n$ is sufficiently large.

##### 2.The algebraic multiplicity of the spectral radius of a hypertree

**Authors:**Lixiang Chen, Changjiang Bu

**Abstract:** It is well-known that the spectral radius of a connected uniform hypergraph is an eigenvalue of the hypergraph. However, its algebraic multiplicity remains unknown. In this paper, we use the Poisson Formula and matching polynomials to determine the algebraic multiplicity of the spectral radius of a uniform hypertree.

##### 3.Implementing Hadamard Matrices in SageMath

**Authors:**Matteo Cati, Dmitrii V. Pasechnik

**Abstract:** Hadamard matrices are $(-1, +1)$ square matrices with mutually orthogonal rows. The Hadamard conjecture states that Hadamard matrices of order $n$ exist whenever $n$ is $1$, $2$, or a multiple of $4$. However, no construction is known that works for all values of $n$, and for some orders no Hadamard matrix has yet been found. Given the many practical applications of these matrices, it would be useful to have a way to easily check if a construction for a Hadamard matrix of order $n$ exists, and in case to create it. This project aimed to address this, by implementing constructions of Hadamard and skew Hadamard matrices to cover all known orders less than or equal to $1000$ in SageMath, an open-source mathematical software. Furthermore, we implemented some additional mathematical objects, such as complementary difference sets and T-sequences, which were not present in SageMath but are needed to construct Hadamard matrices. This also allows to verify the correctness of the results given in the literature; within the $n\leq 1000$ range, just one order, $292$, of a skew Hadamard matrix claimed to have a known construction, required a fix.

##### 4.A new sufficient condition for a 2-strong digraph to be Hamiltonian

**Authors:**Samvel Kh. Darbinyan

**Abstract:** In this paper we prove the following new sufficient condition for a digraph to be Hamiltonian: {\it Let $D$ be a 2-strong digraph of order $n\geq 9$. If $n-1$ vertices of $D$ have degrees at least $n+k$ and the remaining vertex has degree at least $n-k-4$, where $k$ is a non-negative integer, then $D$ is Hamiltonian}. This is an extension of Ghouila-Houri's theorem for 2-strong digraphs and is a generalization of an early result of the author (DAN Arm. SSR (91(2):6-8, 1990). The obtained result is best possible in the sense that for $k=0$ there is a digraph of order $n=8$ (respectively, $n=9$) with the minimum degree $n-4=4$ (respectively, with the minimum $n-5=4$) whose $n-1$ vertices have degrees at least $n-1$, but it is not Hamiltonian. We also give a new sufficient condition for a 3-strong digraph to be Hamiltonian-connected.

##### 5.Spectral radius and k-factor-critical graphs

**Authors:**Sizhong Zhou, Zhiren Sun, Yuli Zhang

**Abstract:** For a nonnegative integer $k$, a graph $G$ is said to be $k$-factor-critical if $G-Q$ admits a perfect matching for any $Q\subseteq V(G)$ with $|Q|=k$. In this article, we prove spectral radius conditions for the existence of $k$-factor-critical graphs. Our result generalises one previous result on perfect matchings of graphs. Furthermore, we claim that the bounds on spectral radius in Theorem 3.1 are sharp.

##### 6.On some series involving the binomial coefficients $\binom{3n}{n}$

**Authors:**Kunle Adegoke, Robert Frontczak, Taras Goy

**Abstract:** Using a simple transformation, we obtain much simpler forms for some series involving binomial coefficients $\binom{3n}n$ derived by Necdet Batir. New evaluations are given; and connections with Fibonacci numbers and the golden ratio are established. Finally, we derive some Fibonacci and Lucas series involving the reciprocals of $\binom{3n}{n}$.

##### 7.Extracting Mergers and Projections of Partitions

**Authors:**Swastik Kopparty, Vishvajeet N

**Abstract:** We study the problem of extracting randomness from somewhere-random sources, and related combinatorial phenomena: partition analogues of Shearer's lemma on projections. A somewhere-random source is a tuple $(X_1, \ldots, X_t)$ of (possibly correlated) $\{0,1\}^n$-valued random variables $X_i$ where for some unknown $i \in [t]$, $X_i$ is guaranteed to be uniformly distributed. An $extracting$ $merger$ is a seeded device that takes a somewhere-random source as input and outputs nearly uniform random bits. We study the seed-length needed for extracting mergers with constant $t$ and constant error. We show: $\cdot$ Just like in the case of standard extractors, seedless extracting mergers with even just one output bit do not exist. $\cdot$ Unlike the case of standard extractors, it $is$ possible to have extracting mergers that output a constant number of bits using only constant seed. Furthermore, a random choice of merger does not work for this purpose! $\cdot$ Nevertheless, just like in the case of standard extractors, an extracting merger which gets most of the entropy out (namely, having $\Omega$ $(n)$ output bits) must have $\Omega$ $(\log n)$ seed. This is the main technical result of our work, and is proved by a second-moment strengthening of the graph-theoretic approach of Radhakrishnan and Ta-Shma to extractors. In contrast, seed-length/output-length tradeoffs for condensing mergers (where the output is only required to have high min-entropy), can be fully explained by using standard condensers. Inspired by such considerations, we also formulate a new and basic class of problems in combinatorics: partition analogues of Shearer's lemma. We show basic results in this direction; in particular, we prove that in any partition of the $3$-dimensional cube $[0,1]^3$ into two parts, one of the parts has an axis parallel $2$-dimensional projection of area at least $3/4$.

##### 8.How Balanced Can Permutations Be?

**Authors:**Gal Beniamini, Nir Lavee, Nati Linial

**Abstract:** A permutation $\pi \in \mathbb{S}_n$ is $k$-balanced if every permutation of order $k$ occurs in $\pi$ equally often, through order-isomorphism. In this paper, we explicitly construct $k$-balanced permutations for $k \le 3$, and every $n$ that satisfies the necessary divisibility conditions. In contrast, we prove that for $k \ge 4$, no such permutations exist. In fact, we show that in the case $k \ge 4$, every $n$-element permutation is at least $\Omega_n(n^{k-1})$ far from being $k$-balanced. This lower bound is matched for $k=4$, by a construction based on the Erd\H{o}s-Szekeres permutation.

##### 9.A combinatorial characterization of $S_2$ binomial edge ideals

**Authors:**Davide Bolognini, Antonio Macchia, Giancarlo Rinaldo, Francesco Strazzanti

**Abstract:** Several algebraic properties of a binomial edge ideal $J_G$ can be interpreted in terms of combinatorial properties of its associated graph $G$. In particular, the so-called cut-point sets of a graph $G$, special sets of vertices that disconnect $G$ in a minimal way, play an important role since they are in bijection with the minimal prime ideals of $J_G$. In this paper we establish the first graph-theoretical characterization of binomial edge ideals $J_G$ satisfying Serre's condition $(S_2)$ by proving that this is equivalent to having $G$ accessible, which means that $J_G$ is unmixed and the cut-point sets of $G$ form an accessible set system. The proof relies on the combinatorial structure of the Stanley-Reisner simplicial complex of a multigraded generic initial ideal of $J_G$, whose facets can be described in terms of cut-point sets. Another key step in the proof consists in proving the equivalence between accessibility and strong accessibility for the collection of cut sets of $G$ with $J_G$ unmixed. This result, interesting on its own, provides the first relevant class of set systems for which the previous two notions are equivalent.

##### 10.Embeddings and hyperplanes of the Lie incidence geometry $A_{n,\{1,n\}}(\mathbb{F})

**Authors:**Antonio Pasini

**Abstract:** In this paper we consider a family of projective embeddings of the geometry $\Gamma = A_{n,\{1,n\}}(F)$ of point-hyperplanes flags of the projective geometry $\Sigma = PG(n,F)$. The natural embedding $\varepsilon_{mathrm{nat}}$ is one of them. It maps every point-hyperplane flag $(p,H)$ of $\Sigma$ onto the vector-line $\langle x\otimes\xi\rangle$, where $x$ is a representative vector of $p$ and $\xi$ is a linear functional describing $H$. The other embeddings have been discovered by Thas and Van Maldeghem (2000) for the case $n = 2$ and later generalized to any $n$ by De Schepper, Schillewaert and Van Maldeghem (2023). They are obtained as twistings of $\varepsilon_{\mathrm{nat}}$ by non-trivial automorphisms of $F$. Explicitly, for $\sigma\in Aut(F)\setminus\{\mathrm{id}_F\}$, the twisting $\varepsilon_\sigma$ of $\varepsilon_{\mathrm{nat}}$ by $\sigma$ maps $(p,H)$ onto $\langle x\sigma\otimes \xi\rangle$. We shall prove that, when $|Aut(F)| > 1$ a geometric hyperplane $\cal H$ of $\Gamma$ arises from $\varepsilon_{\mathrm{nat}}$ and one of its twistings or from two distinct twistings of $\varepsilon_{\mathrm{nat}}$ if and only if ${\cal H} = \{(p,H)\in \Gamma \mid p\in A \mbox{ or } a \in H\}$ for a possibly non-incident point-hyperplane pair $(a,A)$ of $\Sigma$. We call these hyperplanes quasi-singular hyperplanes. With the help of this result we shall prove that if $|Aut(F)| > 1$ then $\Gamma$ admits no absolutely universal embedding.

##### 11.Saturating linear sets of minimal rank

**Authors:**Daniele Bartoli, Martino Borello, Giuseppe Marino

**Abstract:** Saturating sets are combinatorial objects in projective spaces over finite fields that have been intensively investigated in the last three decades. They are related to the so-called covering problem of codes in the Hamming metric. In this paper, we consider the recently introduced linear version of such sets, which is, in turn, related to the covering problem in the rank metric. The main questions in this context are how small the rank of a saturating linear set can be and how to construct saturating linear sets of small rank. Recently, Bonini, Borello, and Byrne provided a lower bound on the rank of saturating linear sets in a given projective space, which is shown to be tight in some cases. In this paper, we provide construction of saturating linear sets meeting the lower bound and we develop a link between the saturating property and the scatteredness of linear sets. The last part of the paper is devoted to show some parameters for which the bound is not tight.

##### 12.Rogers-Ramanujan type identities involving double, triple and quadruple sums

**Authors:**Zhi Li, Liuquan Wang

**Abstract:** We prove a number of new Rogers-Ramanujan type identities involving double, triple and quadruple sums. They were discovered after an extensive search using Maple. The main idea of proofs is to reduce them to some known identities in the literature. This is achieved by direct summation or the constant term method. We also obtain some new single-sum identities as consequences.

##### 13.$3$D Farey graph, lambda lengths and $SL_2$-tilings

**Authors:**Anna Felikson, Oleg Karpenkov, Khrystyna Serhiyenko, Pavel Tumarkin

**Abstract:** We explore a three-dimensional counterpart of the Farey tessellation and its relations to Penner's lambda lengths and $SL_2$-tilings. In particular, we prove a three-dimensional version of Ptolemy relation, and generalise results of Ian Short to classify tame $SL_2$-tilings over Eisenstein integers in terms of pairs of paths in the 3D Farey graph.

##### 1.Fulkerson duality for modulus of spanning trees and partitions

**Authors:**Huy Truong, Pietro Poggi-Corradini

**Abstract:** One of the main properties of modulus on graphs is Fulkerson duality. In this paper, we study Fulkerson duality for spanning tree modulus. We introduce a new notion of Beurling partition, and we identify two important ones, which correspond to the notion of strength and maximum denseness of an arbitrary graph. These special partitions, also give rise to two deflation processes that reveal a hierarchical structure for general graphs. Finally, while Fulkerson duality for spanning tree families can be deduced from a well-known result in combinatorics due to Chopra, we give an alternative approach based on a result of Nash-Williams and Tutte.

##### 2.Some results concerning the valences of (super) edge-magic graphs

**Authors:**Yukio Takahashi, Francesc A. Muntaner-Batle, Rikio Ichishima

**Abstract:** A graph $G$ is called edge-magic if there exists a bijective function $f:V\left(G\right) \cup E\left(G\right)\rightarrow \left\{1, 2, \ldots , \left\vert V\left( G\right) \right\vert +\left\vert E\left( G\right) \right\vert \right\}$ such that $f\left(u\right) + f\left(v\right) + f\left(uv\right)$ is a constant (called the valence of $f$) for each $uv\in E\left( G\right) $. If $f\left(V \left(G\right)\right) =\left\{1, 2, \ldots , \left\vert V\left( G\right) \right\vert \right\}$, then $G$ is called a super edge-magic graph. A stronger version of edge-magic and super edge-magic graphs appeared when the concepts of perfect edge-magic and perfect super edge-magic graphs were introduced. The super edge-magic deficiency $ \mu_{s}\left(G\right)$ of a graph $G$ is defined to be either the smallest nonnegative integer $n$ with the property that $G \cup nK_{1}$ is super edge-magic or $+ \infty$ if there exists no such integer $n$. On the other hand, the edge-magic deficiency $ \mu\left(G\right)$ of a graph $G$ is the smallest nonnegative integer $n$ for which $G\cup nK_{1}$ is edge-magic, being $ \mu\left(G\right)$ always finite. In this paper, the concepts of (super) edge-magic deficiency are generalized using the concepts of perfect (super) edge-magic graphs. This naturally leads to the study of the valences of edge-magic and super edge-magic labelings. We present some general results in this direction and study the perfect (super) edge-magic deficiency of the star $K_{1,n}$.

##### 3.The unicyclic hypergraph with extremal spectral radius

**Authors:**Guanglong Yu, Lin Sun

**Abstract:** For a $hypergraph$ $\mathcal{G}=(V, E)$ consisting of a nonempty vertex set $V=V(\mathcal{G})$ and an edge set $E=E(\mathcal{G})$, its $adjacency$ $matrix$ $\mathcal {A}_{\mathcal{G}}=[(\mathcal {A}_{\mathcal{G}})_{ij}]$ is defined as $(\mathcal {A}_{\mathcal{G}})_{ij}=\sum_{e\in E_{ij}}\frac{1}{|e| - 1}$, where $E_{ij} = \{e \in E \, |\, i, j \in e\}$.The $spectral$ $radius$ of a hypergraph $\mathcal{G}$, denoted by $\rho(\mathcal {G})$, is the maximum modulus among all eigenvalues of $\mathcal {A}_{\mathcal{G}}$. In this paper, among all $k$-uniform ($k\geq 3$) unicyclic hypergraphs with fixed number of vertices, the hypergraphs with the maximum and the second the maximum spectral radius are completely determined, respectively.

##### 4.Collection of Polynomials over Finite Fields Providing Involutary Permutations

**Authors:**P Vanchinathan, Kevinsam B

**Abstract:** For an odd prime power $q$ satisfying $q\equiv 1\pmod 3$ we construct totally $2(q-1) $ permutation polyomials, all giving involutory permutations with exactly $ 1+ \frac{q-1}3$ fixed points. Among them $(q-1)$ polynomials are trinomials, and the rest are 6-term polynomials.

##### 5.On Card guessing games: limit law for one-time riffle shuffle

**Authors:**Markus Kuba, Alois Panholzer

**Abstract:** We consider a card guessing game with complete feedback. A ordered deck of n cards labeled 1 up to n is riffle-shuffled exactly one time. Then, the goal of the game is to maximize the number of correct guesses of the cards, where one after another a single card is drawn from the top, and shown to the guesser until no cards remain. Improving earlier results, we provide a limit law for the number of correct guesses. As a byproduct, we relate the number of correct guesses in this card guessing game to the number of correct guesses under a two-color card guessing game with complete feedback. Using this connection to two-color card guessing, we can also show a limiting distribution result for the first occurrence of a pure luck guess.

##### 6.Homology rings of affine grassmannians and positively multiplicative graphs

**Authors:**Jérémie Guilhot, Cédric Lecouvey, Pierre Tarrago

**Abstract:** Let $\mathfrak{g}$ be an untwisted affine Lie algebra with associated Weyl group $W_a$. To any level 0 weight $\gamma$ we associate a weighted graph $\Gamma_\gamma$ that encodes the orbit of $\gamma$ under the action $W_a$. We show that the graph $\Gamma_\gamma$ encodes the periodic orientation of certain subsets of alcoves in $W_a$ and therefore can be interpreted as an automaton determining the reduced expressions in these subsets. Then, by using some relevant quotients of the homology ring of affine Grassmannians, we show that $\Gamma_\gamma$ is positively multiplicative. This allows us in particular to compute the structure constants of the homology rings using elementary linear algebra on multiplicative graphs. In another direction, the positivity of $\Gamma_\gamma$ yields the key ingredients to study a large class of central random walks on alcoves.

##### 7.New Menger-like dualities in digraphs and applications to half-integral linkages

**Authors:**Victor Campos, Jonas Costa, Raul Lopes, Ignasi Sau

**Abstract:** We present new min-max relations in digraphs between the number of paths satisfying certain conditions and the order of the corresponding cuts. We define these objects in order to capture, in the context of solving the half-integral linkage problem, the essential properties needed for reaching a large bramble of congestion two (or any other constant) from the terminal set. This strategy has been used ad-hoc in several articles, usually with lengthy technical proofs, and our objective is to abstract it to make it applicable in a simpler and unified way. We provide two proofs of the min-max relations, one consisting in applying Menger's Theorem on appropriately defined auxiliary digraphs, and an alternative simpler one using matroids, however with worse polynomial running time. As an application, we manage to simplify and improve several results of Edwards et al. [ESA 2017] and of Giannopoulou et al. [SODA 2022] about finding half-integral linkages in digraphs. Concerning the former, besides being simpler, our proof provides an almost optimal bound on the strong connectivity of a digraph for it to be half-integrally feasible under the presence of a large bramble of congestion two (or equivalently, if the directed tree-width is large, which is the hard case). Concerning the latter, our proof uses brambles as rerouting objects instead of cylindrical grids, hence yielding much better bounds and being somehow independent of a particular topology. We hope that our min-max relations will find further applications as, in our opinion, they are simple, robust, and versatile to be easily applicable to different types of routing problems in digraphs.

##### 8.Genus Permutations and Genus Partitions

**Authors:**Alexander Hock

**Abstract:** For a given permutation or set partition there is a natural way to associate a genus. Counting all permutations or partitions of a fixed genus according to cycle lengths or block sizes, respectively, is the main content of this article. After a variable transformation, the generating series are rational functions with poles located at the ramification points in the new variable. The generating series for any genus is given explicitly for permutations and up to genus 2 for set partitions. Extending the topological structure not just by the genus but also by adding more boundaries, we derive the generating series of non-crossing partitions on the cylinder from known results of non-crossing permutations on the cylinder. Most, but not all, outcomes of this article are special cases of already known results, however they are not represented in this way in the literature, which however seems to be the canonical way. To make the article as accessible as possible, we avoid going into details into the explicit connections to Topological Recursion and Free Probability Theory, where the original motivation came from.

##### 9.On a relationship between the characteristic and matching polynomials of a uniform hypertree

**Authors:**Honghai Li, Li Su, Shaun Fallat

**Abstract:** A hypertree is a connected hypergraph without cycles. Further a hypertree is called an $r$-tree if, additionally, it is $r$-uniform. Note that 2-trees are just ordinary trees. A classical result states that for any 2-tree $T$ with characteristic polynomial $\phi_T(\lambda)$ and matching polynomial $\varphi_T(\lambda)$, then $\phi_T(\lambda)=\varphi_T(\lambda).$ More generally, suppose $\mathcal{T}$ is an $r$-tree of size $m$ with $r\geq2$. In this paper, we extend the above classical relationship to $r$-trees and establish that \[ \phi_{\mathcal{T}}(\lambda)=\prod_{H \sqsubseteq \mathcal{T}}\varphi_{H}(\lambda)^{a_{H}}, \] where the product is over all connected subgraphs $H$ of $\mathcal{T}$, and the exponent $a_{H}$ of the factor $\varphi_{H}(\lambda)$ can be written as \[ a_H=b^{m-e(H)-|\partial(H)|}c^{e(H)}(b-c)^{|\partial(H)|}, \] where $e(H)$ is the size of $H$, $\partial(H)$ is the boundary of $H$, and $b=(r-1)^{r-1}, c=r^{r-2}$. In particular, for $r=2$, the above correspondence reduces to the classical result for ordinary trees. In addition, we resolve a conjecture by Clark-Cooper [{\em Electron. J. Combin.}, 2018] and show that for any subgraph $H$ of an $r$-tree $\mathcal{T}$ with $r\geq3$, $\varphi_H(\lambda)$ divides $\phi_{\mathcal{T}}(\lambda)$, and additionally $\phi_H(\lambda)$ divides $\phi_{\mathcal{T}}(\lambda)$, if either $r\geq 4$ or $H$ is connected when $r=3$. Moreover, a counterexample is given for the case when $H$ is a disconnected subgraph of a 3-tree.

##### 10.Companions to the Andrews-Gordon and Andrews-Bressoud identities, and recent conjectures of Capparelli, Meurman, Primc, and Primc

**Authors:**Matthew C. Russell

**Abstract:** We find bivariate generating functions for the $k=1$ cases of recently conjectured colored partition identities of Capparelli, Meurman, A. Primc, and M. Primc that are slight variants of the generating functions for the sum sides of the Andrews-Gordon and Andrews-Bressoud identities. As a consequence, we prove sum-to-product identities for these cases, thus proving the conjectures.

##### 11.On the Extremal Functions of Acyclic Forbidden 0--1 Matrices

**Authors:**Seth Pettie, Gábor Tardos

**Abstract:** The extremal theory of forbidden 0--1 matrices studies the asymptotic growth of the function $\mathrm{Ex}(P,n)$, which is the maximum weight of a matrix $A\in\{0,1\}^{n\times n}$ whose submatrices avoid a fixed pattern $P\in\{0,1\}^{k\times l}$. This theory has been wildly successful at resolving problems in combinatorics, discrete and computational geometry, structural graph theory, and the analysis of data structures, particularly corollaries of the dynamic optimality conjecture. All these applications use acyclic patterns, meaning that when $P$ is regarded as the adjacency matrix of a bipartite graph, the graph is acyclic. The biggest open problem in this area is to bound $\mathrm{Ex}(P,n)$ for acyclic $P$. Prior results have only ruled out the strict $O(n\log n)$ bound conjectured by Furedi and Hajnal. It is consistent with prior results that $\forall P. \mathrm{Ex}(P,n)\leq n\log^{1+o(1)} n$, and also consistent that $\forall \epsilon>0.\exists P. \mathrm{Ex}(P,n) \geq n^{2-\epsilon}$. In this paper we establish a stronger lower bound on the extremal functions of acyclic $P$. Specifically, we give a new construction of relatively dense 0--1 matrices with $\Theta(n(\log n/\log\log n)^t)$ 1s that avoid an acyclic $X_t$. Pach and Tardos have conjectured that this type of result is the best possible, i.e., no acyclic $P$ exists for which $\mathrm{Ex}(P,n)\geq n(\log n)^{\omega(1)}$.

##### 12.A proof of the Etzion-Silberstein conjecture for monotone and MDS-constructible Ferrers diagrams

**Authors:**Alessandro Neri, Mima Stanojkovski

**Abstract:** Ferrers diagram rank-metric codes were introduced by Etzion and Silberstein in 2009. In their work, they proposed a conjecture on the largest dimension of a space of matrices over a finite field whose nonzero elements are supported on a given Ferrers diagram and all have rank lower bounded by a fixed positive integer $d$. Since stated, the Etzion-Silberstein conjecture has been verified in a number of cases, often requiring additional constraints on the field size or on the minimum rank $d$ in dependence of the corresponding Ferrers diagram. As of today, this conjecture still remains widely open. Using modular methods, we give a constructive proof of the Etzion-Silberstein conjecture for the class of strictly monotone Ferrers diagrams, which does not depend on the minimum rank $d$ and holds over every finite field. In addition, we leverage on the last result to also prove the conjecture for the class of MDS-constructible Ferrers diagrams, without requiring any restriction on the field size.

##### 1.A characterization of graphs of radius-$r$ flip-width at most $2$

**Authors:**Yeonsu Chang, Sejin Ko, O-joung Kwon, Myounghwan Lee

**Abstract:** The $r$-flip-width of a graph, for $r\in \mathbb{N}\cup \{\infty\}$, is a graph parameter defined in terms of a variant of the cops and robber game, called a flipper game, and it was introduced by Toru\'{n}czyk [Flip-width: Cops and robber on dense graphs, arXiv:2302.00352]. We prove that for every $r\in (\mathbb{N}\setminus \{1\})\cup \{\infty\}$, the class of graphs of $r$-flip-width at most $2$ is exactly the class of ($C_5$, bull, gem, co-gem)-free graphs, which are known as totally decomposable graphs with respect to bi-joins.

##### 2.Uniform density in matroids, matrices and graphs

**Authors:**Karel Devriendt, Raffaella Mulas

**Abstract:** We give new characterizations for the class of uniformly dense matroids, and we describe applications to graphic and real representable matroids. We show that a matroid is uniformly dense if and only if its base polytope contains a point with constant coordinates, and if and only if there exists a measure on the bases such that every element of the ground set has equal probability to be in a random basis with respect to this measure. As one application, we derive new spectral, structural and classification results for uniformly dense graphic matroids. In particular, we show that connected regular uniformly dense graphs are $1$-tough and thus contain a (near-)perfect matching. As a second application, we show that strictly uniformly dense real representable matroids can be represented by projection matrices with constant diagonal and that they are parametrized by a subvariety of the real Grassmannian.

##### 3.On a particular specialization of monomial symmetric functions

**Authors:**Vincent Brugidou

**Abstract:** Let $m_{\lambda }$ be the monomial symmetric functions with $\lambda $ integer partition of $n=\left| \lambda \right| $. For the specialization of the $q$-deformation of the exponential, we prove that to each $m_{\lambda }$ is accociated a polynomial $J_{\lambda }\left( q\right) $% , whose coefficients belong to $\mathbb{Z}$. $J_{\lambda }$ is a generalization of the case $\lambda =\left( n\right) $ for which $J_{\left( n\right) }=J_{n}$ is the enumerator polynomial of inversion in tree on $n$ vertices. Some relations between $J_{\lambda }$ and $J_{n,r}$ are obtained, these $J_{n,r}$ having been introduced in $\left[ 4\right] $ from a $q$% -analog of certain symmetric functions, and being themselves inversion enumerator polynomials which generalize $J_{n,1}=J_{n}$. From the calculation of $J_{\lambda }$ for $\left| \lambda \right| \leq 6$, we conjecture that the coefficients of each $J_{\lambda }$\ are strictly positive and log-concave. As a consequence of Huh's works on the $h$-vector of matroid complex (Theorem 3 of $\left[ 7\right] $), it is shown that the coefficients of \ all $J_{n,r}$ are strictly positive and log-concave, which gives a second argument for these conjectures. We prove that the last $n-1$ coefficients of $J_{\lambda }$ are proportional to the first $n-1$ coefficients of column $n-r-1$ of Pascal's triangle, $r$ being the length of $\lambda $. This is a third argument to state the conjectures since the log-concavity of these columns are well known. The calculation of $J_{\left( 3,2,1\right) }$ shows the existence of a obstacle, if one wants to prove the conjectures by application of the theorem of Huh, quoted above.

##### 4.Connectivity of 2-distance graphs

**Authors:**S. H. Jafari, S. R. Musawi

**Abstract:** For a simple graph $G$, the $2$-distance graph, $D_2(G)$, is a graph with the vertex set $V(G)$ and two vertices are adjacent if and only if their distance is $2$ in the graph $G$. In this paper, we characterize all graphs with connected 2-distance graph. For graphs with diameter 2, we prove that $D_2(G)$ is connected if and only if $G$ has no spanning complete bipartite subgraphs. For graphs with a diameter greater than 2, we define a maximal Fine set and by contracting $G$ on these subsets, we get a new graph $\hat G$ such that $D_2(G)$ is connected if and only if $D_2(\hat G)$ is connected. Especially, $D_2(G)$ is disconnected if and only if $\hat G$ is bipartite.

##### 5.The Primitive Eulerian polynomial

**Authors:**Jose Bastidas, Christophe Hohlweg, Franco Saliola

**Abstract:** We introduce the Primitive Eulerian polynomial $P_{\cal A}(z)$ of a central hyperplane arrangement ${\cal A}$. It is a reparametrization of its cocharacteristic polynomial. Previous work of the first author implicitly show that, for simplicial arrangements, $P_{\cal A}(z)$ has nonnegative coefficients. For reflection arrangements of type A and B, the same work interprets the coefficients of $P_{\cal A}(z)$ using the (flag)excedance statistic on (signed) permutations. The main result of this article is to provide an interpretation of the coefficients of $P_{\cal A}(z)$ for all simplicial arrangements only using the geometry and combinatorics of ${\cal A}$. This new interpretation sheds more light to the case of reflection arrangements and, for the first time, gives combinatorial meaning to the coefficients of the Primitive Eulerian polynomial of the reflection arrangement of type D. In type B, we find a connection between the Primitive Eulerian polynomial and the $1/2$-Eulerian polynomial of Savage and Viswanathan (2012). We present some real-rootedness results and conjectures for $P_{\cal A}(z)$.

##### 6.Quadratic embedding constants of graphs: Bounds and distance spectra

**Authors:**Projesh Nath Choudhury, Raju Nandi

**Abstract:** The quadratic embedding constant (QEC) of a finite, simple, connected graph $G$ is the maximum of the quadratic form of the distance matrix of $G$ on the subset of the unit sphere orthogonal to the all-ones vector. The study of these QECs was motivated by the classical work of Schoenberg on quadratic embedding of metric spaces [Ann. of Math., 1935] and [Trans. Amer. Math. Soc., 1938]. In this article, we provide sharp upper and lower bounds for the QEC of trees. We next explore the relation between distance spectra and quadratic embedding constants of graphs - and show two further results: $(i)$ We show that the quadratic embedding constant of a graph is zero if and only if its second largest distance eigenvalue is zero. $(ii)$ We identify a new subclass of nonsingular graphs whose QEC is the second largest distance eigenvalue. Finally, we show that the QEC of the cluster of an arbitrary graph $G$ with either a complete or star graph can be computed in terms of the QEC of $G$. As an application of this result, we provide new families of examples of graphs of QE class.

##### 1.Combinatorics of semi-toric degenerations of Schubert varieties in type C

**Authors:**Naoki Fujita, Yuta Nishiyama

**Abstract:** An approach to Schubert calculus is to realize Schubert classes as concrete combinatorial objects such as Schubert polynomials. Using the polytope ring of the Gelfand-Tsetlin polytopes, Kiritchenko-Smirnov-Timorin realized each Schubert class as a sum of reduced Kogan faces. The first named author introduced a generalization of reduced Kogan faces to symplectic Gelfand-Tsetlin polytopes using a semi-toric degeneration of a Schubert variety, and extended the result of Kiritchenko-Smirnov-Timorin to type C case. In this paper, we introduce a combinatorial model to this type C generalization using a kind of pipe dream with self-crossings. As an application, we prove that the type C generalization can be constructed by skew mitosis operators.

##### 2.An FPT Algorithm for Splitting a Necklace Among Two Thieves

**Authors:**Michaela Borzechowski, Patrick Schnider, Simon Weber

**Abstract:** It is well-known that the 2-Thief-Necklace-Splitting problem reduces to the discrete Ham Sandwich problem. In fact, this reduction was crucial in the proof of the PPA-completeness of the Ham Sandwich problem [Filos-Ratsikas and Goldberg, STOC'19]. Recently, a variant of the Ham Sandwich problem called $\alpha$-Ham Sandwich has been studied, in which the point sets are guaranteed to be well-separated [Steiger and Zhao, DCG'10]. The complexity of this search problem remains unknown, but it is known to lie in the complexity class UEOPL [Chiu, Choudhary and Mulzer, ICALP'20]. We define the analogue of this well-separability condition in the necklace splitting problem -- a necklace is $n$-separable, if every subset $A$ of the $n$ types of jewels can be separated from the types $[n]\setminus A$ by at most $n$ separator points. By the reduction to the Ham Sandwich problem it follows that this version of necklace splitting has a unique solution. We furthermore provide two FPT algorithms: The first FPT algorithm solves 2-Thief-Necklace-Splitting on $(n-1+\ell)$-separable necklaces with $n$ types of jewels and $m$ total jewels in time $2^{O(\ell\log\ell)}+m^2$. In particular, this shows that 2-Thief-Necklace-Splitting is polynomial-time solvable on $n$-separable necklaces. Thus, attempts to show hardness of $\alpha$-Ham Sandwich through reduction from the 2-Thief-Necklace-Splitting problem cannot work. The second FPT algorithm tests $(n-1+\ell)$-separability of a given necklace with $n$ types of jewels in time $2^{O(\ell^2)}\cdot n^4$. In particular, $n$-separability can thus be tested in polynomial time, even though testing well-separation of point sets is coNP-complete [Bergold et al., SWAT'22].

##### 3.Randomly perturbed digraphs also have bounded-degree spanning trees

**Authors:**Patryk Morawski, Kalina Petrova

**Abstract:** We show that a randomly perturbed digraph, where we start with a dense digraph $D_\alpha$ and add a small number of random edges to it, will typically contain a fixed orientation of a bounded degree spanning tree. This answers a question posed by Araujo, Balogh, Krueger, Piga and Treglown and generalizes the corresponding result for randomly perturbed graphs by Krivelevich, Kwan and Sudakov. More specifically, we prove that there exists a constant $c = c(\alpha, \Delta)$ such that if $T$ is an oriented tree with maximum degree $\Delta$ and $D_\alpha$ is an $n$-vertex digraph with minimum semidegree $\alpha n$, then the graph obtained by adding $cn$ uniformly random edges to $D_\alpha$ will contain $T$ with high probability.

##### 4.A new variant of the Erdős-Gyárfás problem on $K_{5}$

**Authors:**Gennian Ge, Zixiang Xu, Yixuan Zhang

**Abstract:** Motivated by an extremal problem on graph-codes that links coding theory and graph theory, Alon recently proposed a question aiming to find the smallest number $t$ such that there is an edge coloring of $K_{n}$ by $t$ colors with no copy of given graph $H$ in which every color appears an even number of times. When $H=K_{4}$, the question of whether $n^{o(1)}$ colors are enough, was initially emphasized by Alon. Through modifications to the coloring functions originally designed by Mubayi, and Conlon, Fox, Lee and Sudakov, the question of $K_{4}$ has already been addressed. Expanding on this line of inquiry, we further study this new variant of the generalized Ramsey problem and provide a conclusively affirmative answer to Alon's question concerning $K_{5}$.

##### 5.Boolean intervals in the weak order of $\mathfrak{S}_n$

**Authors:**Jennifer Elder, Pamela E. Harris, Jan Kretschmann, J. Carlos Martínez Mori

**Abstract:** Let $\mathfrak{S}_n$ denote the symmetric group and let $W(\mathfrak{S}_n)$ denote the weak order of $\mathfrak{S}_n$. Through a surprising connection to a subset of parking functions, which we call \emph{unit Fubini rankings}, we provide a complete characterization and enumeration for the total number of Boolean intervals in $W(\mathfrak{S}_n)$ and the total number of Boolean intervals of rank $k$ in $W(\mathfrak{S}_n)$. Furthermore, for any $\pi\in\mathfrak{S}_n$, we establish that the number of Boolean intervals in $W(\mathfrak{S}_n)$ with minimal element $\pi$ is a product of Fibonacci numbers. We conclude with some directions for further study.

##### 6.Matroid Products in Tropical Geometry

**Authors:**Nicholas Anderson

**Abstract:** Symmetric powers of matroids were first introduced by Lovasz and Mason in the 1970s, where it was shown that not all matroids admit higher symmetric powers. Since these initial findings, the study of matroid symmetric powers has remained largely unexplored. In this paper, we establish an equivalence between valuated matroids with arbitrarily large symmetric powers and tropical linear spaces that appear as the variety of a tropical ideal. In establishing this equivalence, we additionally show that all tropical linear spaces are connected through codimension one. These results provide additional geometric and algebraic connections to the study of matroid symmetric powers that we hope will lend future insight to the study of tropical ideals and their varieties, and vice versa.

##### 1.Normal 5-edge-coloring of some snarks superpositioned by Flower snarks

**Authors:**Jelena Sedlar, Riste Škrekovski

**Abstract:** An edge e is normal in a proper edge-coloring of a cubic graph G if the number of distinct colors on four edges incident to e is 2 or 4: A normal edge-coloring of G is a proper edge-coloring in which every edge of G is normal. The Petersen Coloring Conjecture is equivalent to stating that every bridgeless cubic graph has a normal 5-edge-coloring. Since every 3-edge-coloring of a cubic graph is trivially normal, it is suficient to consider only snarks to establish the conjecture. In this paper, we consider a class of superpositioned snarks obtained by choosing a cycle C in a snark G and superpositioning vertices of C by one of two simple supervertices and edges of C by superedges Hx;y, where H is any snark and x; y any pair of nonadjacent vertices of H: For such superpositioned snarks, two suficient conditions are given for the existence of a normal 5-edge-coloring. The first condition yields a normal 5-edge-coloring for all hypohamiltonian snarks used as superedges, but only for some of the possible ways of connecting them. In particular, since the Flower snarks are hypohamiltonian, this consequently yields a normal 5-edge-coloring for many snarks superpositioned by the Flower snarks. The second sufficient condition is more demanding, but its application yields a normal 5-edge-colorings for all superpositions by the Flower snarks. The same class of snarks is considered in [S. Liu, R.-X. Hao, C.-Q. Zhang, Berge{Fulkerson coloring for some families of superposition snarks, Eur. J. Comb. 96 (2021) 103344] for the Berge-Fulkerson conjecture. Since we established that this class has a Petersen coloring, this immediately yields the result of the above mentioned paper.

##### 2.Extrema of local mean and local density in a tree

**Authors:**Ruoyu Wang

**Abstract:** Given a tree T, one can define the local mean at some subtree S to be the average order of subtrees containing S. It is natural to ask which subtree of order k achieves the maximal/minimal local mean among all the subtrees of the same order and what properties it has. We call such subtrees k-maximal subtrees. Wagner and Wang showed in 2016 that a 1- maximal subtree is a vertex of degree 1 or 2. This paper shows that for any integer k = 1, . . . , |T| , a k-maximal subtree has at most one leaf whose degree is greater than 2 and at least one leaf whose degree is at most 2. Furthermore, we show that a k-maximal subtree has a leaf of degree greater than 2 only when all its other leaves are leaves in T as well. In the second part, this paper introduces the local density as a normalization of local means, for the sake of comparing subtrees of different orders, and shows that the local density at subtree S is lower-bounded by 1/2 with equality if and only if S contains the core of T. On the other hand, local density can be arbitrarily close to 1.

##### 3.Minimal Face Numbers for Volume Rigidity

**Authors:**Jack Southgate

**Abstract:** Maxwell counts give bounds on the numbers of edges required for Euclidean bar-joint rigidity in $\mathbb{R}^d$ in terms of the number of vertices, as well as their sparsity. In this paper, we give the necessary lower bounds on each of the face numbers of a simplex required for that simplex to be $d$-volume rigid in $\mathbb{R}^d$, using a recent application of algebraic combinatorial techniques to $d$-volume rigidity. In order to do so, we prove some facts about the $d$-volume rigidity matroid, noting combinatorial characterisations when $d = 1$ and $2$. Finally, we state a $d$-volume rigidity Vertex Removal Lemma and give an improved statement using our lower bounds.

##### 4.Planar Turán number of the 7-cycle

**Authors:**Ruilin Shi, Zach Walsh, Xingxing Yu

**Abstract:** The $\textit{planar Tur\'an number}$ $\textrm{ex}_{\mathcal P}(n,H)$ of a graph $H$ is the maximum number of edges in an $n$-vertex planar graph without $H$ as a subgraph. Let $C_{\ell}$ denote the cycle of length $\ell$. The planar Tur\'an number $\textrm{ex}_{\mathcal P}(n,C_{\ell})$ behaves differently for $\ell\le 10$ and for $\ell\ge 11$, and it is known when $\ell \in \{3,4,5,6\}$. We prove that $\textrm{ex}_{\mathcal P}(n,C_7) \le \frac{18n}{7} - \frac{48}{7}$ for all $n > 38$, and show that equality holds for infinitely many integers $n$.

##### 5.Positive del Pezzo Geometry

**Authors:**Nick Early, Alheydis Geiger, Marta Panizzut, Bernd Sturmfels, Claudia He Yun

**Abstract:** Real, complex, and tropical algebraic geometry join forces in a new branch of mathematical physics called positive geometry. We develop the positive geometry of del Pezzo surfaces and their moduli spaces, viewed as very affine varieties. Their connected components are derived from polyhedral spaces with Weyl group symmetries. We study their canonical forms and scattering amplitudes, and we solve the likelihood equations.

##### 1.Counting occurrences of patterns in permutations

**Authors:**Andrew R Conway, Anthony J Guttmann

**Abstract:** We develop a new, powerful method for counting elements in a {\em multiset.} As a first application, we use this algorithm to study the number of occurrences of patterns in a permutation. For patterns of length 3 there are two Wilf classes, and the general behaviour of these is reasonably well-known. We slightly extend some of the known results in that case, and exhaustively study the case of patterns of length 4, about which there is little previous knowledge. For such patterns, there are seven Wilf classes, and based on extensive enumerations and careful series analysis, we have conjectured the asymptotic behaviour for all classes. Finally, we investigate a proposal of Blitvi\'c and Steingr\'imsson as to the range of a parameter for which a particular generating function formed from the occurrence sequences is itself a Stieltjes moment sequence.

##### 2.A constructive solution to OP with a large cycle

**Authors:**Tommaso Traetta

**Abstract:** For every $2$-regular graph $F$ of order $v$, the Oberwolfach problem $OP(F)$ asks whether there is a $2$-factorization of $K_v$ ($v$ odd) or $K_v$ minus a $1$-factor ($v$ even) into copies of $F$. Posed by Ringel in 1967 and extensively studied ever since, this problem is still open. In this paper we construct solutions to $OP(F)$ whenever $F$ contains a cycle of length greater than an explicit lower bound. Our constructions combine the amalgamation-detachment technique with methods aimed at building $2$-factorizations with an automorphism group having a nearly-regular action on the vertex-set.

##### 3.Advancements in Research Mathematics through AI: A Framework for Conjecturing

**Authors:**Randy Davila

**Abstract:** This paper introduces a general framework for computer-based conjecture generation, particularly those conjectures that mathematicians might deem substantial and elegant. We describe our approach and demonstrate its effectiveness by providing examples of its application in producing publishable research and unexpected mathematical insights. We anticipate that our discussion of computer-assisted mathematical conjecturing will catalyze further research into this area and encourage the development of more advanced techniques than the ones presented herein.

##### 4.A telescopic proof of Cayley's formula

**Authors:**Guillaume Chapuy, Guillem Perarnau

**Abstract:** We give a short proof of the fact that the number of labelled trees on $n$ vertices is $n^{n-2}$. Although many short proofs are known, we have not seen this one before.

##### 5.On boundedness of zeros of the independence polynomial of tor

**Authors:**David de Boer, Pjotr Buys, Han Peters, Guus Regts

**Abstract:** We study boundedness of zeros of the independence polynomial of tori for sequences of tori converging to the integer lattice. We prove that zeros are bounded for sequences of balanced tori, but unbounded for sequences of highly unbalanced tori. Here balanced means that the size of the torus is at most exponential in the shortest side length, while highly unbalanced means that the longest side length of the torus is super exponential in the product over the other side lengths cubed. We discuss implications of our results to the existence of efficient algorithms for approximating the independence polynomial on tori.

##### 6.The binomial random graph is a bad inducer

**Authors:**Vishesh Jain, Marcus Michelen

**Abstract:** For a finite graph $F$ and a value $p \in [0,1]$, let $I(F,p)$ denote the largest $y$ for which there is a sequence of graphs of edge density approaching $p$ so that the induced $F$-density of the sequence approaches $y$. In this short note, we show that for all $F$ on at least three vertices and $p \in (0,1)$, the binomial random graph $G(n,p)$ has induced $F$-density strictly less than $I(F,p).$ This provides a negative answer to a problem posed by Liu, Mubayi and Reiher.

##### 7.A book proof of the middle levels theorem

**Authors:**Torsten Mütze

**Abstract:** We give a short constructive proof for the existence of a Hamilton cycle in the subgraph of the $(2n+1)$-dimensional hypercube induced by all vertices with exactly $n$ or $n+1$ many 1s.

##### 8.Combinatorial Fiedler Theory and Graph Partition

**Authors:**Enide Andrade, Geir Dahl

**Abstract:** Partition problems in graphs are extremely important in applications, as shown in the Data science and Machine learning literature. One approach is spectral partitioning based on a Fiedler vector, i.e., an eigenvector corresponding to the second smallest eigenvalue $a(G)$ of the Laplacian matrix $L_G$ of the graph $G$. This problem corresponds to the minimization of a quadratic form associated with $L_G$, under certain constraints involving the $\ell_2$-norm. We introduce and investigate a similar problem, but using the $\ell_1$-norm to measure distances. This leads to a new parameter $b(G)$ as the optimal value. We show that a well-known cut problem arises in this approach, namely the sparsest cut problem. We prove connectivity results and different bounds on this new parameter, relate to Fiedler theory and show explicit expressions for $b(G)$ for trees. We also comment on an $\ell_{\infty}$-norm version of the problem.

##### 9.Flattened Stirling Permutations

**Authors:**Adam Buck, Jennifer Elder, Azia A. Figueroa, Pamela E. Harris, Kimberly Harry, Anthony Simpson

**Abstract:** Recall that a Stirling permutation is a permutation on the multiset $\{1,1,2,2,\ldots,n,n\}$ such that any numbers appearing between repeated values of $i$ must be greater than $i$. We call a Stirling permutation ``flattened'' if the leading terms of maximal chains of ascents (called runs) are in weakly increasing order. Our main result establishes a bijection between flattened Stirling permutations and type $B$ set partitions of $\{0,\pm1,\pm2,\ldots,\pm (n-1)\}$, which are known to be enumerated by the Dowling numbers, and we give an independent proof of this fact. We also determine the maximal number of runs for any flattened Stirling permutation, and we enumerate flattened Stirling permutations with a small number of runs or with two runs of equal length. We conclude with some conjectures and generalizations worthy of future investigation.

##### 10.Lucky Cars and the Quicksort Algorithm

**Authors:**Pamela E. Harris, Jan Kretschmann, J. Carlos Martínez Mori

**Abstract:** Quicksort is a classical divide-and-conquer sorting algorithm. It is a comparison sort that makes an average of $2(n+1)H_n - 4n$ comparisons on an array of size $n$ ordered uniformly at random, where $H_n = \sum_{i=1}^n\frac{1}{i}$ is the $n$th harmonic number. Therefore, it makes $n!\left[2(n+1)H_n - 4n\right]$ comparisons to sort all possible orderings of the array. In this article, we prove that this count also enumerates the parking preference lists of $n$ cars parking on a one-way street with $n$ parking spots resulting in exactly $n-1$ lucky cars (i.e., cars that park in their preferred spot). For $n\geq 2$, both counts satisfy the second order recurrence relation $ f_n=2nf_{n-1}-n(n-1)f_{n-2}+2(n-1)! $ with $f_0=f_1=0$.

##### 1.On several problems of defective choosability

**Authors:**Jie Ma, Rongxing Xu, Xuding Zhu

**Abstract:** Given positive integers $p \ge k$, and a non-negative integer $d$, we say a graph $G$ is $(k,d,p)$-choosable if for every list assignment $L$ with $|L(v)|\geq k$ for each $v \in V(G)$ and $|\bigcup_{v\in V(G)}L(v)| \leq p$, there exists an $L$-coloring of $G$ such that each monochromatic subgraph has maximum degree at most $d$. In particular, $(k,0,k)$-choosable means $k$-colorable, $(k,0,+\infty)$-choosable means $k$-choosable and $(k,d,+\infty)$-choosable means $d$-defective $k$-choosable. This paper proves that there are 3-colorable planar graphs that are not $1$-defective $3$-choosable, there are 1-defective 3-choosable graphs that are not 4-choosable, and for any positive integers $\ell \geq k \geq 3$, and non-negative integer $d$, there are $(k,d, \ell)$-choosable graphs that are not $(k,d , \ell+1)$-choosable. These results answer questions asked by \v{S}krekovski [Combin. Probab. Comput. 8, 3(1999), 293-299], Wang and Xu [SIAM J. Discrete Math. 27, 4(2013), 2020-2037], and Kang [J. Graph Theory 73, 3(2013), 342-353], respectively. Our construction of $(k,d, \ell)$-choosable but not $(k,d , \ell+1)$-choosable graphs generalizes the construction of Kr\'{a}l' and Sgall in [J. Graph Theory 49, 3(2005), 177-186] for the case $d=0$.

##### 2.Cyclic relative difference families with block size four and their applications

**Authors:**Chenya Zhao, Binwei Zhao, Yanxun Chang, Tao Feng, Xiaomiao Wang, Menglong Zhang

**Abstract:** Given a subgroup $H$ of a group $(G,+)$, a $(G,H,k,1)$ difference family (DF) is a set $\mathcal F$ of $k$-subsets of $G$ such that $\{f-f':f,f'\in F, f\neq f',F\in \mathcal F\}=G\setminus H$. Let $g\mathbb Z_{gh}$ is the subgroup of order $h$ in $\mathbb Z_{gh}$ generated by $g$. A $(\mathbb Z_{gh},g\mathbb Z_{gh},k,1)$-DF is called cyclic and written as a $(gh,h,k,1)$-CDF. This paper shows that for $h\in\{2,3,6\}$, there exists a $(gh,h,4,1)$-CDF if and only if $gh\equiv h\pmod{12}$, $g\geq 4$ and $(g,h)\not\in\{(9,3),(5,6)\}$. As a corollary, it is shown that a 1-rotational S$(2,4,v)$ exists if and only if $v\equiv4\pmod{12}$ and $v\neq 28$. This solves the long-standing open problem on the existence of a 1-rotational S$(2,4,v)$. As another corollary, we establish the existence of an optimal $(v,4,1)$-optical orthogonal code with $\lfloor(v-1)/12\rfloor$ codewords for any positive integer $v\equiv 1,2,3,4,6\pmod{12}$ and $v\neq 25$. We also give applications of our results to cyclic group divisible designs with block size four and optimal cyclic $3$-ary constant-weight codes with weight four and minimum distance six.

##### 1.Bounds on the genus for 2-cell embeddings of prefix-reversal graphs

**Authors:**Saúl A. Blanco, Charles Buehrle

**Abstract:** In this paper, we provide bounds for the genus of the pancake graph $\mathbb{P}_n$, burnt pancake graph $\mathbb{BP}_n$, and undirected generalized pancake graph $\mathbb{P}_m(n)$. Our upper bound for $\mathbb{P}_n$ is sharper than the previously-known bound, and the other bounds presented are the first of their kind. Our proofs are constructive and rely on finding an appropriate rotation system (also referred to in the literature as Edmonds' permutation technique) where certain cycles in the graphs we consider become boundaries of regions of a 2-cell embedding. A key ingredient in the proof of our bounds for the genus $\mathbb{P}_n$ and $\mathbb{BP}_n$ is a labeling algorithm of their vertices that allows us to implement rotation systems to bound the number of regions of a 2-cell embedding of said graphs.

##### 2.Free paths of arrangements of hyperplanes

**Authors:**Takuro Abe, Toru Yamaguchi

**Abstract:** We study the free path problem, i.e., if we are given two free arrangements of hyperplanes, then we can connect them by free arrangements or not. We prove that if an arrangement $\mathcal{A}$ and $\mathcal{A} \setminus \{H,L\}$ are free, then at least one of two among them is free. When $\mathcal{A}$ is in the three dimensional arrangement, we show a stronger statement.

##### 3.On extremal trees with respect to Mostar index

**Authors:**Fazal Hayat, Shou-Jun Xu

**Abstract:** For a given graph $G$, the Mostar index $Mo(G)$ is defined as the sum of absolute values of the differences between $n_u(e)$ and $n_v(e)$ over all edges $e=uv$ of $G$, where $n_u(e)$ and $n_v(e)$ are respectively, the number of vertices of $G$ lying closer to $u$ than to $v$ and the number of vertices of $G$ lying closer to $v$ than to $u$. We determine the unique graphs having largest and smallest Mostar index over all trees of order $n$ with fixed parameters such as number of odd vertices, number of vertices of degree two and the number of pendent paths of fixed length. Moreover, we identify the unique graph that minimizes the Mostar index from the class of all trees of order $n$ with fixed number of branch vertices.

##### 4.Shellable simplicial complex and switching rook polynomial of frame polyominoes

**Authors:**Rizwan Jahangir, Francesco Navarra

**Abstract:** Let $\mathcal{P}$ be a frame polyomino, a new kind of non-simple polyomino. In this paper we study the $h$-polynomial of $K[\mathcal{P}]$ in terms of the switching rook polynomial of $\mathcal{P}$ using the shellable simplicial complex $\Delta(\mathcal{P})$ attached to $\mathcal{P}$. We provide a suitable shelling order for $\Delta(\mathcal{P})$ in relation to a new combinatorial object, which we call a step of a facet, and we define a bijection between the set of the canonical configuration of $k$ rooks in $\mathcal{P}$ and the facets of $\Delta(\mathcal{P})$ with $k$ steps. Finally we use a famous combinatorial result, due to McMullen and Walkup, about the $h$-vector of a shellable simplicial complex to interpret the $h$-polynomial of $K[\mathcal{P}]$ as the switching rook polynomial of $\mathcal{P}$.

##### 5.A remark on continued fractions for permutations and D-permutations with a weight $-1$ per cycle

**Authors:**Bishal Deb, Alan D. Sokal

**Abstract:** We show that very simple continued fractions can be obtained for the ordinary generating functions enumerating permutations or D-permutations with a large number of independent statistics, when each cycle is given a weight $-1$. The proof is based on a simple lemma relating the number of cycles modulo 2 to the numbers of fixed points, cycle peaks (or cycle valleys), and crossings.

##### 6.Focusing on gap lengths in the noisy violinist chip-firing problem

**Authors:**Leonid Ryvkin

**Abstract:** We encode the states in the noisy violinist chip-firing problem into a sequence of gap lengths, and use this gap-focused perspective to reprove certain statements about final states of flat clusterons.

##### 7.Invariant systems of weighted representatives

**Authors:**Anton A. Klyachko, Mikhail S. Terekhov

**Abstract:** It is known that, if removing some $n$ edges from a graph $\Gamma$ destroys all subgraphs isomorphic to a given finite graph $K$, then all subgraphs isomorphic to $K$ can be destroyed by removing at most $|E(K)|\cdot n$ edges, which form a set invariant with respect to all automorphisms of $\Gamma$. We construct the first examples of (connected) graphs $K$ for which this estimate is not sharp. Our arguments are based on a ``weighted analogue'' of an earlier known estimate for the cost of symmetry.

##### 8.Normality of $k$-Matching Polytopes of Bipartite Graphs

**Authors:**Juan Camilo Torres

**Abstract:** The $k$-matching polytope of a graph is the convex hull of all its matchings of a given size $k$ when they are considered as indicator vectors. In this paper, we prove that the $k$-matching polytope of a bipartite graph is normal, that is, every integer point in its $t$-dilate is the sum of $t$ integers points of the original polytope. This generalizes the known fact that Birkhoff polytopes are normal. As a preliminary result, we prove that for bipartite graphs the $k$-matching polytope is equal to the fractional $k$-matching polytope, having thus the $H$-representation of the polytope. This generalizes the Birkhoff-Von Neumann Theorem which establish that every doubly stochastic matrix can be written as a convex combination of permutation matrices.

##### 9.Group irregularity strength of disconnected graphs

**Authors:**Sylwia Cichacz, Barbara Krupińska

**Abstract:** We investigate the \textit{group irregular strength} $(s_g(G))$ of graphs, i.e the smallest value of $s$ such that for any Abelian group $\Gamma$ of order $s$ exists a function $g\colon E(G) \rightarrow \Gamma$ such that sums of edge labels at every vertex is distinct. We give results for bound and exact values of $(s_g(G))$ for graphs without small stars as components.

##### 10.Experimenting with Discrete Dynamical Systems

**Authors:**George Spahn, Doron Zeilberger

**Abstract:** We demonstrate the power of Experimental Mathematics and Symbolic Computation to study intriguing problems on rational difference equations, studied extensively by Difference Equations giants, Saber Elaydi and Gerry Ladas (and their students and collaborators). In particular we rigorously prove some fascinating conjectures made by Amal Amleh and Gerry Ladas back in 2000. For other conjectures we are content with semi-rigorous proofs. We also extend the work of Emilie Purvine (formerly Hogan) and Doron Zeilberger for rigorously and semi-rigorously proving global asymptotic stability of arbitrary rational difference equations (with positive coefficients), and more generally rational transformations of the positive orthant of $R^k$ into itself

##### 11.Extremal bounds for pattern avoidance in multidimensional 0-1 matrices

**Authors:**Jesse Geneson, Shen-Fu Tsai

**Abstract:** A 0-1 matrix $M$ contains another 0-1 matrix $P$ if some submatrix of $M$ can be turned into $P$ by changing any number of $1$-entries to $0$-entries. $M$ is $\mathcal{P}$-saturated where $\mathcal{P}$ is a family of 0-1 matrices if $M$ avoids every element of $\mathcal{P}$ and changing any $0$-entry of $M$ to a $1$-entry introduces a copy of some element of $\mathcal{P}$. The extremal function ex$(n,\mathcal{P})$ and saturation function sat$(n,\mathcal{P})$ are the maximum and minimum possible weight of an $n\times n$ $\mathcal{P}$-saturated 0-1 matrix, respectively, and the semisaturation function ssat$(n,P)$ is the minimum possible weight of an $n\times n$ $\mathcal{P}$-semisaturated 0-1 matrix $M$, i.e., changing any $0$-entry in $M$ to a $1$-entry introduces a new copy of some element of $\mathcal{P}$. We study these functions of multidimensional 0-1 matrices. We give upper bounds on parameters of minimally non-$O(n^{d-1})$ $d$-dimensional 0-1 matrices, generalized from minimally nonlinear 0-1 matrices in two dimensions, and we show the existence of infinitely many minimally non-$O(n^{d-1})$ $d$-dimensional 0-1 matrices with all dimensions of length greater than $1$. For any positive integers $k,d$ and integer $r\in[0,d-1]$, we construct a family of $d$-dimensional 0-1 matrices with both extremal function and saturation function exactly $kn^r$ for sufficiently large $n$. We show that no family of $d$-dimensional 0-1 matrices has saturation function strictly between $O(1)$ and $\Theta(n)$ and we construct a family of $d$-dimensional 0-1 matrices with bounded saturation function and extremal function $\Omega(n^{d-\epsilon})$ for any $\epsilon>0$. Up to a constant multiplicative factor, we fully settle the problem of characterizing the semisaturation function of families of $d$-dimensional 0-1 matrices, which we prove to always be $\Theta(n^r)$ for some integer $r\in[0,d-1]$.

##### 1.Properties of Villarceau Torus

**Authors:**Paul Manuel

**Abstract:** Villarceau torus is a discrete graph theory model of spiral torus which is called Helical Toroidal Electron Model in Physics. It also represents the double stranded helix model of DNA. The spiral torus or toroidal helix in Physics and Molecular Biology is continuous whereas the Villarceau torus is discrete. The main contribution of the paper is the identification of cycles which are equivalent to toroidal helix in spiral torus. In addition, the poloidal revolution and the toroidal revolutions of Villarceau torus are computed. The paper identifies some striking differences between ring torus and Villarceau torus. It is proved that Villarceau torus does not admit any convex edgecuts and convex cycles other than 4-cycles. The convex cut method is extended to graphs which do not admit convex edgecuts. Using this technique, the network distance (Wiener Index) is computed, Also an optimal congestion-balanced routing for Villarceau torus is designed.

##### 2.A Note on Hamiltonian Cycles in Digraphs with Large Degrees

**Authors:**Samvel Kh. Darbinyan

**Abstract:** In this note we prove: {\it Let $D$ be a 2-strong digraph of order $n$ such that its $n-1$ vertices have degrees at least $n+k$ and the remaining vertex $z$ has degree at least $n-k-4$, where $k$ is a positive integer. If $D$ contains a cycle of length at least $n-k-2$ passing through $z$, then $D$ is Hamiltonian}.

##### 3.Matching Fields in Macaulay2

**Authors:**Oliver Clarke

**Abstract:** This article introduces the package MatchingFields for Macaulay2 and highlights some open problems. A matching field is a combinatorial object whose data encodes a candidate toric degeneration of a Grassmannian or partial flag variety of type A. Each coherent matching field is associated to a certain maximal cone of the respective tropical variety. The MatchingFields package provides methods to construct matching fields along with their rings, ideals, polyhedra and matroids. The package also supplies methods to test whether a matching field is coherent, linkage and gives rise to a toric degeneration.

##### 4.Minimum $\ell$-degree thresholds for rainbow perfect matching in $k$-uniform hypergraphs

**Authors:**Jie You

**Abstract:** Given $n\in k\mathbb{N}$ elements set $V$ and $k$-uniform hypergraphs $\mathcal{H}_1,\ldots,\mathcal{H}_{n/k}$ on $V$. A rainbow perfect matching is a collection of pairwise disjoint edges $E_1\in \mathcal{H}_1,\ldots,E_{n/k}\in \mathcal{H}_{n/k}$ such that $E_1\cup\cdots\cup E_{n/k}=V$. In this paper, we determine the minimum $\ell$-degree condition that guarantees the existence of a rainbow perfect matching for sufficiently large $n$ and $\ell\geq k/2$.

##### 5.Affine stresses, inverse systems, and reconstruction problems

**Authors:**Satoshi Murai, Isabella Novik, Hailun Zheng

**Abstract:** A conjecture of Kalai asserts that for $d\geq 4$, the affine type of a prime simplicial $d$-polytope $P$ can be reconstructed from the space of affine $2$-stresses of $P$. We prove this conjecture for all $d\geq 5$. We also prove the following generalization: for all pairs $(i,d)$ with $2\leq i\leq \lceil \frac d 2\rceil-1$, the affine type of a simplicial $d$-polytope $P$ that has no missing faces of dimension $\geq d-i+1$ can be reconstructed from the space of affine $i$-stresses of $P$. A consequence of our proofs is a strengthening of the Generalized Lower Bound Theorem: for any simplicial $(d-1)$-sphere $\Delta$ and $1\leq k\leq \lceil\frac{d}{2}\rceil-1$, $g_k(\Delta)$ is at least as large as the number of missing $(d-k)$-faces of $\Delta$; furthermore, for $1\leq k\leq \lfloor\frac{d}{2}\rfloor-1$, equality holds if and only if $\Delta$ is $k$-stacked. Finally, we show that for $d\geq 4$, any simplicial $d$-polytope $P$ that has no missing faces of dimension $\geq d-1$ is redundantly rigid, that is, for each edge $e$ of $P$, there exists an affine $2$-stress on $P$ with a non-zero value on $e$.

##### 6.Clustered coloring of (path+2K_1)-free graphs on surfaces

**Authors:**Zdeněk Dvořák

**Abstract:** Esperet and Joret proved that planar graphs with bounded maximum degree are 3-colorable with bounded clustering. Liu and Wood asked whether the conclusion holds with the assumption of the bounded maximum degree replaced by assuming that no two vertices have many common neighbors. We answer this question in positive, in the following stronger form: Let P''_t be the complete join of two isolated vertices with a path on t vertices. For any surface Sigma, a subgraph-closed class of graphs drawn on Sigma is 3-choosable with bounded clustering if and only if there exists t such that P''_t does not belong to the class.

##### 7.Constructing generalized Heffter arrays via near alternating sign matrices

**Authors:**Lorenzo Mella, Tommaso Traetta

**Abstract:** Let $S$ be a subset of a group $G$ (not necessarily abelian) such that $S\,\cap -S$ is empty or contains only elements of order $2$, and let $\mathbf{h}=(h_1,\ldots, h_m)\in \mathbb{N}^m$ and $\mathbf{k}=(k_1, \ldots, k_n)\in \mathbb{N}^n$. A generalized Heffter array GHA$^{\lambda}_S(m, n; \mathbf{h}, \mathbf{k})$ over $G$ is an $m\times n$ matrix $A=(a_{ij})$ such that: the $i$-th row (resp. $j$-th column) of $A$ contains exactly $h_i$ (resp. $k_j$) nonzero elements, and the list $\{a_{ij}, -a_{ij}\mid a_{ij}\neq 0\}$ equals $\lambda$ times the set $S\,\cup\, -S$. We speak of a zero sum (resp. nonzero sum) GHA if each row and each column of $A$ sums to zero (resp. a nonzero element), with respect to some ordering. In this paper, we use near alternating sign matrices to build both zero and nonzero sum GHAs, over cyclic groups, having the further strong property of being simple. In particular, we construct zero sum and simple GHAs whose row and column weights are congruent to $0$ modulo $4$. Furthermore, we build nonzero sum GHA$^{\lambda}_S(m, n; \mathbf{h}, \mathbf{k})$ over an arbitrary group $G$ whenever $S$ contains enough noninvolutions, thus extending previous nonconstructive results where $\pm S = G\setminus H$ for some subgroup $H$ of $G$. Finally, we describe how GHAs can be used to build orthogonal decompositions and biembeddings of Cayley graphs (over groups not necessarily abelian) onto orientable surfaces.

##### 8.Lower General Position Sets in Graphs

**Authors:**Gabriele Di Stefano, Sandi Klavžar, Aditi Krishnakumar, James Tuite, Ismael Yero

**Abstract:** A subset $S$ of vertices of a graph $G$ is a \emph{general position set} if no shortest path in $G$ contains three or more vertices of $S$. In this paper, we generalise a problem of M. Gardner to graph theory by introducing the \emph{lower general position number} $\gp ^-(G)$ of $G$, which is the number of vertices in a smallest maximal general position set of $G$. We show that ${\rm gp}^-(G) = 2$ if and only if $G$ contains a universal line and determine this number for several classes of graphs, including Kneser graphs $K(n,2)$, line graphs of complete graphs, and Cartesian and direct products of two complete graphs. We also prove several realisation results involving the lower general position number, the general position number and the geodetic number, and compare it with the lower version of the monophonic position number. We provide a sharp upper bound on the size of graphs with given lower general position number. Finally we demonstrate that the decision version of the lower general position problem is NP-complete.

##### 9.On Permutation Trinomials of the type $X^{q^2-q+1}+AX^{q^2}+BX$ over $\mathbb{F}_{q^3}$

**Authors:**Daniele Bartoli, Francesco Ghiandoni

**Abstract:** Necessary and sufficient conditions on $A,B\in \mathbb{F}_{q^3}^*$ for $f(X)=X^{q^2-q+1}+AX^{q^2}+BX$ being a permutation polynomial of $\mathbb{F}_{q^3}$ are investigated via a connection with algebraic varieties over finite fields.

##### 1.Connectivity of graphs that do not have the edge-Erdős-Pósa property

**Authors:**Raphael Steck

**Abstract:** I show that we can assume graphs that do not have the edge-Erd\H{o}s-P\'{o}sa property to be connected. Then I strengthen this result to $2$-connectivity under the additional assumptions of a minor-closed property and a generic counterexample.

##### 2.Directed cycles with zero weight in $\mathbb{Z}_p^k$

**Authors:**Shoham Letzter, Natasha Morrison

**Abstract:** For a finite abelian group $A$, define $f(A)$ to be the minimum integer such that for every complete digraph $\Gamma$ on $f$ vertices and every map $w:E(\Gamma) \rightarrow A$, there exists a directed cycle $C$ in $\Gamma$ such that $\sum_{e \in E(C)}w(e) = 0$. The study of $f(A)$ was initiated by Alon and Krivelevich (2021). In this article, we prove that $f(\mathbb{Z}_p^k) = O(pk (\log k)^2)$, where $p$ is prime, with an improved bound of $O(k \log k)$ when $p = 2$. These bounds are tight up to a factor which is polylogarithmic in $k$.

##### 3.Synchronizing random automata through repeated 'a' inputs

**Authors:**Anders Martinsson

**Abstract:** In a recent article by Chapuy and Perarnau, it was shown that a uniformly chosen automaton on $n$ states with a $2$-letter alphabet has a synchronizing word of length $O(\sqrt{n}\log n)$ with high probability. In this note, we give a new simplified proof of a slightly weaker version of this statement. Our proof is based on two properties of random automata. First, by repeating a fixed character from the alphabet sufficiently many times in a row, the number of possible states reduces to, in expectation, $O(\sqrt{n})$. Second, with high probability, each pair of states can be synchronized by a word of length $O(\log n)$.

##### 4.Subcubic graphs of large treewidth do not have the edge-Erdős-Pósa property

**Authors:**Raphael Steck, Henning Bruhn

**Abstract:** We show that subcubic graphs of treewidth at least $2500$ do not have the edge-Erd\H{o}s-P\'{o}sa property.

##### 5.Decomposition of an Integrally Convex Set into a Minkowski Sum of Bounded and Conic Integrally Convex Sets

**Authors:**Kazuo Murota, Akihisa Tamura

**Abstract:** Every polyhedron can be decomposed into a Minkowski sum (or vector sum) of a bounded polyhedron and a polyhedral cone. This paper establishes similar statements for some classes of discrete sets in discrete convex analysis, such as integrally convex sets, L-natural-convex sets, and M-natural-convex sets.

##### 6.Mostar index and bounded maximum degree

**Authors:**Michael A. Henning, Johannes Pardey, Dieter Rautenbach, Florian Werner

**Abstract:** Do\v{s}li\'{c} et al. defined the Mostar index of a graph $G$ as $Mo(G)=\sum\limits_{uv\in E(G)}|n_G(u,v)-n_G(v,u)|$, where, for an edge $uv$ of $G$, the term $n_G(u,v)$ denotes the number of vertices of $G$ that have a smaller distance in $G$ to $u$ than to $v$. For a graph $G$ of order $n$ and maximum degree at most $\Delta$, we show $Mo(G)\leq \frac{\Delta}{2}n^2-(1-o(1))c_{\Delta}n\log(\log(n)),$ where $c_{\Delta}>0$ only depends on $\Delta$ and the $o(1)$ term only depends on $n$. Furthermore, for integers $n_0$ and $\Delta$ at least $3$, we show the existence of a $\Delta$-regular graph of order $n$ at least $n_0$ with $Mo(G)\geq \frac{\Delta}{2}n^2-c'_{\Delta}n\log(n),$ where $c'_{\Delta}>0$ only depends on $\Delta$.

##### 7.Countable ultrahomogeneous graphs on two imprimitive color classes

**Authors:**Sofia Brenner, Irene Heinrich

**Abstract:** We classify the countable ultrahomogeneous 2-vertex-colored graphs in which the color classes are imprimitive, i.e., up to complementation they form disjoint unions of cliques. This generalizes work by Jenkinson, Lockett and Truss as well as Rose on ultrahomogeneous $n$-graphs. As the key aspect in such a classification, we identify a concept called piecewise ultrahomogeneity. We prove that there are two specific graphs whose occurrence essentially dictates whether a graph is piecewise ultrahomogeneous, and we exploit this fact to prove the classification.

##### 8.Cycles with many chords

**Authors:**Nemanja Draganić, Abhishek Methuku, David Munhá Correia, Benny Sudakov

**Abstract:** How many edges in an $n$-vertex graph will force the existence of a cycle with as many chords as it has vertices? Almost 30 years ago, Chen, Erd\H{o}s and Staton considered this question and showed that any $n$-vertex graph with $2n^{3/2}$ edges contains such a cycle. We significantly improve this old bound by showing that $\Omega(n\log^8n)$ edges are enough to guarantee the existence of such a cycle. Our proof exploits a delicate interplay between certain properties of random walks in almost regular expanders. We argue that while the probability that a random walk of certain length in an almost regular expander is self-avoiding is very small, one can still guarantee that it spans many edges (and that it can be closed into a cycle) with large enough probability to ensure that these two events happen simultaneously.

##### 9.On the cross-product conjecture for the number of linear extensions

**Authors:**Swee Hong Chan, Igor Pak, Greta Panova

**Abstract:** We prove a weak version of the cross--product conjecture: ${F}(k+1,\ell) {F}(k,\ell+1) \geq (\frac12+\varepsilon) {F}(k,\ell) {F}(k+1,\ell+1)$, where ${F}(k,\ell)$ is the number of linear extensions for which the values at fixed elements $x,y,z$ are $k$ and $\ell$ apart, respectively, and where $\varepsilon>0$ depends on the poset. We also prove the converse inequality and disprove the {generalized cross--product conjecture}. The proofs use geometric inequalities for mixed volumes and combinatorics of words.

##### 10.Chip-firing and critical groups of signed graphs

**Authors:**Matthew Cho, Anton Dochtermann, Ryota Inagaki, Suho Oh, Dylan Snustad, Bailee Zacovic

**Abstract:** A \textit{signed graph} $G_\phi$ consists of a graph $G$ along with a function $\phi$ that assigns a positive or negative weight to each edge. The reduced signed Laplacian matrix $L_{G_\phi}$ gives rise to a natural notion of chip-firing on $G_\phi$, as well as a critical group ${\mathcal K}(G_\phi)$. Here a negative edge designates an adversarial relationship, so that firing a vertex incident to such an edge leads to a loss of chips at both endpoints. We study chip-firing on signed graphs, employing the theory of chip-firing on invertible matrices introduced by Guzm\'an and Klivans. %Here valid chip configurations are given by the lattice points of a certain cone determined by $G_\phi$ and the underlying graph $G$. This gives rise to notions of \textit{critical} as well as \textit{$z$-superstable} configurations, both of which are counted by the determinant of $L_{G_\phi}$. We establish general results regarding these configurations, focusing on efficient methods of verifying the underlying properties. We then study the critical group of signed graphs in the context of vertex switching and Smith normal forms. We use this to compute the critical groups for various classes of signed graphs including signed cycles, wheels, complete graphs, and fans. In the process we generalize a number of results from the literature.

##### 1.New Explicit Constant-Degree Lossless Expanders

**Authors:**Louis Golowich

**Abstract:** We present a new explicit construction of onesided bipartite lossless expanders of constant degree, with arbitrary constant ratio between the sizes of the two vertex sets. Our construction is simpler to state and analyze than the prior construction of Capalbo, Reingold, Vadhan, and Wigderson (2002). We construct our lossless expanders by imposing the structure of a constant-sized lossless expander "gadget" within the neighborhoods of a large bipartite spectral expander; similar constructions were previously used to obtain the weaker notion of unique-neighbor expansion. Our analysis simply consists of elementary counting arguments and an application of the expander mixing lemma.

##### 2.Stability for hyperplane covers

**Authors:**Shagnik Das, Valjakas Djaljapayan, Yen-chi Roger Lin, Wei-Hsuan Yu

**Abstract:** An almost $k$-cover of the hypercube $Q^n = \{0,1\}^n$ is a collection of hyperplanes that avoids the origin and covers every other vertex at least $k$ times. When $k$ is large with respect to the dimension $n$, Clifton and Huang asymptotically determined the minimum possible size of an almost $k$-cover. Central to their proof was an extension of the LYM inequality, concerning a weighted count of hyperplanes. In this paper we completely characterise the hyperplanes of maximum weight, showing that there are $\binom{2n-1}{n}$ such planes. We further provide stability, bounding the weight of all hyperplanes that are not of maximum weight. These results allow us to effectively shrink the search space when using integer linear programming to construct small covers, and as a result we are able to determine the exact minimum size of an almost $k$-cover of $Q^6$ for most values of $k$. We further use the stability result to improve the Clifton--Huang lower bound for infinitely many choices of $k$ in every sufficiently large dimension $n$.

##### 3.Distance Magic Labeling of Generalised Mycielskian Graphs

**Authors:**Ravindra Pawar, Tarkehswar Singh

**Abstract:** In this paper, we have studied the distance magic labelling of Generalised Mycielskian of a few families of graphs.

##### 4.Outerplane bipartite graphs with isomorphic resonance graphs

**Authors:**Simon Brezovnik, Zhongyuan Che, Niko Tratnik, Petra Žigert Pleteršek

**Abstract:** We present novel results related to isomorphic resonance graphs of 2-connected outerplane bipartite graphs. As the main result, we provide a structure characterization for 2-connected outerplane bipartite graphs with isomorphic resonance graphs. Moreover, two additional characterizations are expressed in terms of resonance digraphs and via local structures of inner duals of 2-connected outerplane bipartite graphs, respectively.

##### 5.A sufficient condition for the existence of fractional $(g,f,n)$-critical covered graphs

**Authors:**Jie Wu

**Abstract:** In data transmission networks, the availability of data transmission is equivalent to the existence of the fractional factor of the corresponding graph which is generated by the network. Research on the existence of fractional factors under specific network structures can help scientists design and construct networks with high data transmission rates. A graph $G$ is called a fractional $(g,f)$-covered graph if for any $e\in E(G)$, $G$ admits a fractional $(g,f)$-factor covering $e$. A graph $G$ is called a fractional $(g,f,n)$-critical covered graph if after removing any $n$ vertices of $G$, the resulting graph of $G$ is a fractional $(g,f)$-covered graph. In this paper, we verify that if a graph $G$ of order $p$ satisfies $p\geq\frac{(a+b-1)(a+b-2)+(a+d)n+1}{a+d}$, $\delta(G)\geq\frac{(b-d-1)p+(a+d)n+a+b+1}{a+b-1}$ and $\delta(G)>\frac{(b-d-2)p+2\alpha(G)+(a+d)n+1}{a+b-2}$, then $G$ is a fractional $(g,f,n)$-critical covered graph, where $g,f:V(G)\rightarrow Z^{+}$ be two functions such that $a\leq g(x)\leq f(x)-d\leq b-d$ for all $x\in V(G)$, which is a generalization of Zhou's previous result [S. Zhou, Some new sufficient conditions for graphs to have fractional $k$-factors, International Journal of Computer Mathematics 88(3)(2011)484--490].

##### 6.Shifted Hankel determinants of Catalan numbers and related results II: Backward shifts

**Authors:**Johann Cigler

**Abstract:** By prepending zeros to a given sequence Hankel determinants of backward shifts of this sequence become meaningful. We obtain some results for the sequences of Catalan numbers and of some numbers and polynomials which are related to Catalan numbers and propose conjectures for sequences of convolution powers of Catalan numbers.

##### 7.On the $α$-index of minimally $k$-(edge-)connected graphs for small $k$

**Authors:**Jiayu Lou, Ligong Wang, Ming Yuan

**Abstract:** Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of vertex degrees of $G$. For any real $\alpha \in [0,1]$, Nikiforov defined the $A_\alpha$-matrix of a graph $G$ as $A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G)$. The largest eigenvalue of $A_\alpha(G)$ is called the $\alpha$-index or the $A_\alpha$-spectral radius of $G$. A graph is minimally $k$-(edge)-connected if it is $k$-(edge)-connected and deleting any arbitrary chosen edge always leaves a graph which is not $k$-(edge)-connected. In this paper, we characterize the minimally 2-edge-connected graphs and minimally 3-connected graph with given order having the maximum $\alpha$-index for $\alpha \in [\frac{1}{2},1)$, respectively.

##### 8.Tight lower bounds for anti-concentration of Rademacher sums and Tomaszewski's counterpart problem

**Authors:**Lawrence Hollom, Julien Portier

**Abstract:** In this paper we prove that $\mathbb{P}(|X| \geq \sqrt{\text{Var}(X)}) \geq 7/32$ for every finite Rademacher sum $X$, confirming a conjecture by Hitczenko and Kwapie{\'n} from 1994, and improving upon results from Burkholder, Oleszkiewicz, and Dvo\v{r}\'ak and Klein. Moreover we fully determine the function $f(y)= \inf_X \mathbb{P}(|X| \geq y\sqrt{\text{Var}(X)})$ where the $\inf$ is taken over all finite Rademacher sums $X$, confirming a conjecture by Lowther and giving a partial answer to a question by Keller and Klein.

##### 9.On the intersection spectrum of $\operatorname{PSL}_2(q)$

**Authors:**Angelot Behajaina, Roghayeh Maleki, Andriaherimanana Sarobidy Razafimahatratra

**Abstract:** Given a group $G$ and a subgroup $H \leq G$, a set $\mathcal{F}\subset G$ is called $H$\emph{-intersecting} if for any $g,g' \in \mathcal{F}$, there exists $xH \in G/H$ such that $gxH=g'xH$. The \emph{intersection density} of the action of $G$ on $G/H$ by (left) multiplication is the rational number $\rho(G,H)$, equal to the maximum ratio $\frac{|\mathcal{F}|}{|H|}$, where $\mathcal{F} \subset G$ runs through all $H$-intersecting sets of $G$. The \emph{intersection spectrum} of the group $G$ is then defined to be the set $$ \sigma(G) := \left\{ \rho(G,H) : H\leq G \right\}. $$ It was shown by Bardestani and Mallahi-Karai [{\it J. Algebraic Combin.}, 42(1):111-128, 2015] that if $\sigma(G) = \{1\}$, then $G$ is necessarily solvable. The natural question that arises is, therefore, which rational numbers larger than $1$ belong to $\sigma(G)$, whenever $G$ is non-solvable. In this paper, we study the intersection spectrum of the linear group $\operatorname{PSL}_2(q)$. It is shown that $2 \in \sigma\left(\operatorname{PSL}_2(q)\right)$, for any prime power $q\equiv 3 \pmod 4$. Moreover, when $q\equiv 1 \pmod 4$, it is proved that $\rho(\operatorname{PSL}_2(q),H)=1$, for any odd index subgroup $H$ (containing $\mathbb{F}_q$) of the Borel subgroup (isomorphic to $\mathbb{F}_q\rtimes \mathbb{Z}_{\frac{q-1}{2}}$) consisting of all upper triangular matrices.

##### 10.Subsequence frequency in binary words

**Authors:**Krishna Menon, Anurag Singh

**Abstract:** The numbers we study in this paper are of the form $B_{n, p}(k)$, which is the number of binary words of length $n$ that contain the word $p$ (as a subsequence) exactly $k$ times. Our motivation comes from the analogous study of pattern containment in permutations. In our first set of results, we obtain explicit expressions for $B_{n, p}(k)$ for small values of $k$. We then focus on words $p$ with at most $3$ runs and study the maximum number of occurrences of $p$ a word of length $n$ can have. We also study the internal zeros in the sequence $(B_{n, p}(k))_{k \geq 0}$ for fixed $n$ and discuss the unimodality and log-concavity of such sequences.

##### 11.On the edge-density of the Brownian co-graphon and common ancestors of pairs in the CRT

**Authors:**Guillaume Chapuy

**Abstract:** Bassino et al. (arXiv:1907.08517) have shown that uniform random co-graphs (graphs without induced $P_4$) of size $n$ converge to a certain non-deterministic graphon. The edge-density of this graphon is a random variable $\Lambda \in [0,1]$ whose first moments have been computed by these authors. The first purpose of this note is to observe that, in fact, these moments can be computed by a simple recurrence relation. The problem leads us to the following question of independent interest: given $k$ i.i.d. uniform pairs of points $(X_1,Y_1),$ $\dots$, $(X_k,Y_k)$ in the Brownian CRT, what is the size $S_k$ of the set $\{X_i\wedge Y_i, i=1\dots k\}$ formed by their pairwise last common ancestors? We show that ${S_k}\sim c{\sqrt{k}\log k}$ in probability, with $c=(\sqrt{2\pi})^{-1}$. The method to establish the recurrence relation is reminiscent of Janson's computation of moments of the Wiener index of (large) random trees. The logarithm factor in the convergence result comes from the estimation of Riemann sums in which summands are weighted by the integer divisor function -- such sums naturally occur in the problem. We have not been able to analyse the asymptotics of moments of $\Lambda$ directly from the recurrence relation, and in fact our study of $S_k$ is independent from it. Several things remain to be done, in particular we only scratch the question of large deviations of $S_k$, and the precise asymptotics of moments of $\Lambda$ is left open.

##### 12.Adaptive Monte Carlo Search for Conjecture Refutation in Graph Theory

**Authors:**Valentino Vito, Lim Yohanes Stefanus

**Abstract:** Graph theory is an interdisciplinary field of study that has various applications in mathematical modeling and computer science. Research in graph theory depends on the creation of not only theorems but also conjectures. Conjecture-refuting algorithms attempt to refute conjectures by searching for counterexamples to those conjectures, often by maximizing certain score functions on graphs. This study proposes a novel conjecture-refuting algorithm, referred to as the adaptive Monte Carlo search (AMCS) algorithm, obtained by modifying the Monte Carlo tree search algorithm. Evaluated based on its success in finding counterexamples to several graph theory conjectures, AMCS outperforms existing conjecture-refuting algorithms. The algorithm is further utilized to refute six open conjectures, two of which were chemical graph theory conjectures formulated by Liu et al. in 2021 and four of which were formulated by the AutoGraphiX computer system in 2006. Finally, four of the open conjectures are strongly refuted by generalizing the counterexamples obtained by AMCS to produce a family of counterexamples. It is expected that the algorithm can help researchers test graph-theoretic conjectures more effectively.

##### 1.Braids act on configurations of lines

**Authors:**Vassily Olegovich Manturov

**Abstract:** Similar pictures appear in various branches of mathematics. Sometimes this similarity gives rise to deep theorems. Mentioning such a similarity between hexagonal tilings, cubes in 3-space, configurations of lines and braid groups, we prove that braids act on configurations of lines.

##### 2.Vertex-shellings of Euclidean Oriented Matroids

**Authors:**Winfried Hochstättler, Michael Wilhelmi

**Abstract:** We prove that a lexicographical extension of a Euclidean oriented matroid remains Euclidean. Based on that result we show that in a Euclidean oriented matroid program there exists a topological sweep inducing a recursive atom-ordering (a shelling of the cocircuits) of the tope cell of the feasible region. We extend that sweep and obtain also a vertex-shelling of the whole oriented matroid and finally describe some connections to the notion of stackable zontope tilings and to a counterexample of a conjecture of A. Mandel.

##### 3.Indecomposable combinatorial games

**Authors:**Michael Fisher, Neil A. McKay, Rebecca Milley, Richard J. Nowakowski, Carlos P. Santos

**Abstract:** In Combinatorial Game Theory, short game forms are defined recursively over all the positions the two players are allowed to move to. A form is decomposable if it can be expressed as a disjunctive sum of two forms with smaller birthday. If there are no such summands, then the form is indecomposable. The main contribution of this document is the characterization of the indecomposable nimbers and the characterization of the indecomposable numbers. More precisely, a nimber is indecomposable if and only if its size is a power of two, and a number is indecomposable if and only if its absolute value is less or equal than one.

##### 4.Colouring random graphs: Tame colourings

**Authors:**Annika Heckel, Konstantinos Panagiotou

**Abstract:** Given a graph G, a colouring is an assignment of colours to the vertices of G so that no two adjacent vertices are coloured the same. If all colour classes have size at most t, then we call the colouring t-bounded, and the t-bounded chromatic number of G, denoted by $\chi_t(G)$, is the minimum number of colours in such a colouring. Every colouring of G is then $\alpha(G)$-bounded, where $\alpha(G)$ denotes the size of a largest independent set. We study colourings of the random graph G(n,1/2) and of the corresponding uniform random graph G(n,m) with $m=\left \lfloor \frac 12 {n \choose 2} \right \rfloor$. We show that $\chi_t(G(n,m))$ is maximally concentrated on at most two explicit values when $t = \alpha(G(n,m))-2$. This behaviour stands in stark contrast to the normal chromatic number, which was recently shown not to be concentrated on any sequence of intervals of length $n^{1/2-o(1)}$. Moreover, when $t = \alpha(G_{n, 1/2})-1$, we find an explicit interval of length $n^{0.99}$ that contains $\chi_t(G(n, 1/2))$ with high probability. Both results have profound consequences: the former is at the core of the tantalising Zigzag Conjecture on the distribution of the chromatic number of G(n, 1/2) and justifies one of its main hypotheses, while the latter is an important ingredient in the proof of a non-concentration result by the first author and Oliver Riordan for $\chi(G(n, 1/2))$ which is conjectured to be optimal. Our two aforementioned results are consequences of a more general statement. We consider a class of colourings that we call tame and which fulfil some natural, albeit technical, conditions. We provide tight bounds for the probability of existence of such colourings, and then we apply these bounds to the case of t-bounded colourings. As a further consequence of our general result, we show two-point concentration of the equitable chromatic number of G(n,m).

##### 1.Generalized stepwise transmission irregular graphs

**Authors:**Yaser Alizadeh, Sandi Klavžar, Zohre Molaee

**Abstract:** The transmission ${\rm Tr}_G(u)$ of a vertex $u$ of a connected graph $G$ is the sum of distances from $u$ to all other vertices. $G$ is a stepwise transmission irregular (STI) graph if $|{\rm Tr}_G(u) - {\rm Tr}_G(v)|= 1$ holds for any edge $uv\in E(G)$. In this paper, generalized STI graphs are introduced as the graphs $G$ such that for some $k\ge 1$ we have $|{\rm Tr}_G(u) - {\rm Tr}_G(v)|= k$ for any edge $uv$ of $G$. It is proved that generalized STI graphs are bipartite and that as soon as the minimum degree is at least $2$, they are 2-edge connected. Among the trees, the only generalized STI graphs are stars. The diameter of STI graphs is bounded and extremal cases discussed. The Cartesian product operation is used to obtain highly connected generalized STI graphs. Several families of generalized STI graphs are constructed.

##### 2.The Maker-Maker domination game in forests

**Authors:**Eric Duchêne, Arthur Dumas, Nacim Oijid, Aline Parreau, Eric Rémila

**Abstract:** We study the Maker-Maker version of the domination game introduced in 2018 by Duch\^ene et al. Given a graph, two players alternately claim vertices. The first player to claim a dominating set of the graph wins. As the Maker-Breaker version, this game is PSPACE-complete on split and bipartite graphs. Our main result is a linear time algorithm to solve this game in forests. We also give a characterization of the cycles where the first player has a winning strategy.

##### 3.A dichotomy theorem for $Γ$-switchable $H$-colouring on $m$-edge coloured graphs

**Authors:**Richard Brewster, Arnott Kinder, Gary MacGillivray

**Abstract:** Let $G$ be a graph in which each edge is assigned one of the colours $1, 2, \ldots, m$, and let $\Gamma$ be a subgroup of $S_m$. The operation of switching at a vertex $x$ of $G$ with respect to an element $\pi$ of $\Gamma$ permutes the colours of the edges incident with $x$ according to $\pi$. We investigate the complexity of whether there exists a sequence of switches that transforms a given $m$-edge coloured graph $G$ so that it has a colour-preserving homomorphism to a fixed $m$-edge coloured graph $H$ and give a dichotomy theorem in the case that $\Gamma$ acts transitively.

##### 4.The Lights Out Game on Directed Graphs

**Authors:**T. Elise Dettling, Darren B. Parker

**Abstract:** We study a version of the lights out game played on directed graphs. For a digraph $D$, we begin with a labeling of $V(D)$ with elements of $\mathbb{Z}_k$ for $k \ge 2$. When a vertex $v$ is toggled, the labels of $v$ and any vertex that $v$ dominates are increased by 1 mod $k$. The game is won when each vertex has label 0. We say that $D$ is $k$-Always Winnable (also written $k$-AW) if the game can be won for every initial labeling with elements of $\mathbb{Z}_k$. We prove that all acyclic digraphs are $k$-AW for all $k$, and we reduce the problem of determining whether a graph is $k$-AW to the case of strongly connected digraphs. We then determine winnability for tournaments with a minimum feedback arc set that arc-induces a directed path or directed star digraph.

##### 1.Colouring planar graphs with a precoloured induced cycle

**Authors:**Ajit Diwan

**Abstract:** Let $C$ be a cycle and $f : V(C) \rightarrow \{c_1,c_2,\ldots,c_k\}$ a proper $k$-colouring of $C$ for some $k \ge 4$. We say the colouring $f$ is safe if for any planar graph $G$ in which $C$ is an induced cycle, there exists a proper $k$-colouring $f'$ of $G$ such that $f'(v) = f(v)$ for all $v \in V(C)$. The only safe $4$-colouring is any proper colouring of a triangle. We give a simple necessary condition for a $k$-colouring of a cycle to be safe and conjecture that it is sufficient for all $k \ge 4$. The sufficiency for $k=4$ follows from the four colour theorem and we prove it for $k = 5$, independent of the four colour theorem. We show that a stronger condition is sufficient for all $k \ge 4$. As a consequence, it follows that any proper $k$-colouring of a cycle that uses at most $k-3$ distinct colours is safe. Also, any proper $k$-colouring of a cycle of length at most $2k-5$ that uses at most $k-1$ distinct colours is safe.

##### 2.Powers of Karpelevic arcs and their Sparsest Realising matrices

**Authors:**Priyanka Joshi, Stephen Kirkland, Helena Smigoc

**Abstract:** The region in the complex plane containing the eigenvalues of all stochastic matrices of order n was described by Karpelevic in 1988, and it is since then known as the Karpelevic region. The boundary of the Karpelevic region is the union of disjoint arcs called the Karpelevic arcs. We provide a complete characterization of the Karpelevic arcs that are powers of some other Karpelevic arc. Furthermore, we find the necessary and sufficient conditions for a sparsest stochastic matrix associated with the Karpelevic arc of order n to be a power of another stochastic matrix.

##### 3.Sparse Incidence Geometries and Pebble Game Algorithms

**Authors:**Signe Lundqvist, Tovohery Randrianarisoa, Klara Stokes, Joannes Vermant

**Abstract:** In this paper we define sparsity and tightness of rank 2 incidence geometries, and we develop an algorithm which recognises these properties. We give examples from rigidity theory where such sparsity conditions are of interest. Under certain conditions, this algorithm also allows us to find a maximum size subgeometry which is tight. This work builds on so-called pebble game algorithms for graphs and hypergraphs. The main difference compared to the previously studied hypergraph case is that in this paper, the sparsity and tightness are defined in terms of incidences, and not in terms of edges. This difference makes our algorithm work not only for uniform hypergraphs, but for all hypergraphs.

##### 4.Disproof of a conjecture of Conlon, Fox and Wigderson

**Authors:**Chunchao Fan, Qizhong Lin, Yuanhui Yan

**Abstract:** For graphs $G$ and $H$, the Ramsey number $r(G,H)$ is the smallest positive integer $N$ such that any red/blue edge coloring of the complete graph $K_N$ contains either a red $G$ or a blue $H$. A book $B_n$ is a graph consisting of $n$ triangles all sharing a common edge. Recently, Conlon, Fox and Wigderson (2023) conjecture that for any $0<\alpha<1$, the random lower bound $r(B_{\lceil\alpha n\rceil},B_n)\ge (\sqrt{\alpha}+1)^2n+o(n)$ would not be tight. In other words, there exists some constant $\beta=\beta(\alpha)>0$ such that $r(B_{\lceil\alpha n\rceil},B_n)\ge (\sqrt{\alpha}+1)^2n+\beta n$ for all sufficiently large $n$. This conjecture clearly holds for every $\alpha< 1/6$ from an early result of Nikiforov and Rousseau (2005), i.e., for every $\alpha< 1/6$ and large $n$, $r(B_{\lceil\alpha n\rceil},B_n)=2n+3$. We disprove the conjecture of Conlon et al. (2023). Indeed, we show that the random lower bound is asymptotically tight for every $1/4\leq \alpha\leq 1$. Moreover, we show that for any $1/6\leq \alpha\le 1/4$ and large $n$, $r(B_{\lceil\alpha n\rceil}, B_n)\le\left(\frac 32+3\alpha\right) n+o(n)$, where the inequality is asymptotically tight when $\alpha=1/6$ or $1/4$. We also give a lower bound of $r(B_{\lceil\alpha n\rceil}, B_n)$ for $1/6\le\alpha< \frac{52-16\sqrt{3}}{121}\approx0.2007$, showing that the random lower bound is not tight, i.e., the conjecture of Conlon et al. (2023) holds in this interval.

##### 5.covering simplicial game complex

**Authors:**Neda Shojaee, Morteza M. Rezaii

**Abstract:** In this paper, we introduce a simplicial complex representation for finite non-cooperative games in the strategic form. The covering space of the simplicial game complex is introduced and we show that the covering complex is a powerful tool to find Nash Equilibrium simplices. This representation allows us to model the cost functions of a game as a weight number on a dual vertex of the strategy situation in some stars. It yields a canonical direct sum decomposition of an arbitrary game into three components, as the potential, harmonic and nonstrategic components.

##### 6.A note on the distance and distance signless Laplacian spectral radius of complements of trees

**Authors:**Iswar Mahato, M. Rajesh Kannan

**Abstract:** In this article, we show that the generalized tree shift operation increases the distance spectral radius, distance signless Laplacian spectral radius, and the $D_\alpha$-spectral radius of complements of trees. As a consequence of this result, we correct an ambiguity in the proofs of some of the known results.

##### 7.Quantum computing algorithms for inverse problems on graphs and an NP-complete inverse problem

**Authors:**Joonas Ilmavirta, Matti Lassas, Jinpeng Lu, Lauri Oksanen, Lauri Ylinen

**Abstract:** We consider an inverse problem for a finite graph $(X,E)$ where we are given a subset of vertices $B\subset X$ and the distances $d_{(X,E)}(b_1,b_2)$ of all vertices $b_1,b_2\in B$. The distance of points $x_1,x_2\in X$ is defined as the minimal number of edges needed to connect two vertices, so all edges have length 1. The inverse problem is a discrete version of the boundary rigidity problem in Riemannian geometry or the inverse travel time problem in geophysics. We will show that this problem has unique solution under certain conditions and develop quantum computing methods to solve it. We prove the following uniqueness result: when $(X,E)$ is a tree and $B$ is the set of leaves of the tree, the graph $(X,E)$ can be uniquely determined in the class of all graphs having a fixed number of vertices. We present a quantum computing algorithm which produces a graph $(X,E)$, or one of those, which has a given number of vertices and the required distances between vertices in $B$. To this end we develop an algorithm that takes in a qubit representation of a graph and combine it with Grover's search algorithm. The algorithm can be implemented using only $O(|X|^2)$ qubits, the same order as the number of elements in the adjacency matrix of $(X,E)$. It also has a quadratic improvement in computational cost compared to standard classical algorithms. Finally, we consider applications in theory of computation, and show that a slight modification of the above inverse problem is NP-complete: all NP-problems can be reduced to a discrete inverse problem we consider.

##### 8.Vertex isoperimetry on signed graphs and spectra of non-bipartite Cayley graphs

**Authors:**Chunyang Hu, Shiping Liu

**Abstract:** For a non-bipartite finite Cayley graph, we show the non-trivial eigenvalues of its normalized adjacency matrix lie in the interval $$\left[-1+\frac{ch_{out}^2}{d},1-\frac{Ch_{out}^2}{d}\right],$$ for some absolute constant $c$ and $C$, where $h_{out}$ stands for the outer vertex boundary isoperimetric constant. This improves upon recent obtained estimates aiming at a quantitative version of an observation due to Breuillard, Green, Guralnick and Tao stating that if a non-bipartite finite Cayley graph is an expander then the non-trivial eigenvalues of its normalized adjacency matrix is not only bounded away from $1$ but also bounded away from $-1$. We achieve this by extending the work of Bobkov, Houdr\'e and Tetali on vertex isoperimetry to the setting of signed graphs. Our approach answers positively a recent open question proposed by Moorman, Ralli and Tetali.

##### 9.Twin-width of subdivisions of multigraphs

**Authors:**Jungho Ahn, Debsoumya Chakraborti, Kevin Hendrey, Sang-il Oum

**Abstract:** For each $d\leq3$, we construct a finite set $F_d$ of multigraphs such that for each graph $H$ of girth at least $5$ obtained from a multigraph $G$ by subdividing each edge at least two times, $H$ has twin-width at most $d$ if and only if $G$ has no minor in $F_d$. This answers a question of Berg\'{e}, Bonnet, and D\'{e}pr\'{e}s asking for the structure of graphs $G$ such that each long subdivision of $G$ has twin-width $4$. As a corollary, we show that the $7\times7$ grid has twin-width $4$, which answers a question of Schidler and Szeider.

##### 10.The sum of all width-one tensors

**Authors:**William Q. Erickson, Jan Kretschmann

**Abstract:** This paper generalizes a recent result by the authors concerning the sum of width-one matrices; in the present work, we consider width-one tensors of arbitrary dimensions. A tensor is said to have width 1 if, when visualized as an array, its nonzero entries lie along a path consisting of steps in the directions of the standard coordinate vectors. We prove two different formulas to compute the sum of all width-one tensors with fixed dimensions and fixed sum of (nonnegative integer) components. The first formula is obtained by converting width-one tensors into tuples of one-row semistandard Young tableaux; the second formula, which extracts coefficients from products of multiset Eulerian polynomials, is derived via Stanley-Reisner theory, making use of the EL-shelling of the order complex on the standard basis of tensors.

##### 1.Group connectivity of 3-edge-connected signed graphs

**Authors:**Alejandra Brewer Castano, Jessica McDonald, Kathryn Nurse

**Abstract:** Jaeger, Linial, Payan, and Tarsi introduced the notion of $A$-connectivity for graphs in 1992, and proved a decomposition for cubic graphs from which $A$-connectivity follows for all 3-edge-connected graphs when $|A|\geq 6$. The concept of $A$-connectivity was generalized to signed graphs by Li, Luo, Ma, and Zhang in 2018 and they proved that all 4-edge-connected flow-admissible signed graphs are $A$-connected when $|A|\geq 4$ and $|A|\neq 5$. We prove that all 3-edge-connected flow-admissible signed graphs are $A$-connected when $|A|\geq 6$ and $|A|\neq 7$. Our proof is based on a decomposition that is a signed-graph analogue of the decomposition found by Jaeger et. al, and which may be of independent interest.

##### 2.Connection between Schubert polynomials and top Lascoux polynomials

**Authors:**Tianyi Yu

**Abstract:** Schubert polynomials form a basis of the polynomial ring. This basis and its structure constants have received extensive study. Recently, Pan and Yu initiated the study of top Lascoux polynomials. These polynomials form a basis of a subalgebra of the polynomial ring where each graded piece has finite dimension. This paper connects Schubert polynomials and top Lascoux polynomials via a simple operator. We use this connection to show these two bases share the same structure constants. We also translate several results on Schubert polynomials to top Lascoux polynomials, including combinatorial formulas for their monomial expansions and supports.

##### 3.Local version of Vizing's theorem for multi-graphs

**Authors:**Clinton T. Conley, Jan Grebik, Oleg Pikhurko

**Abstract:** Extending a result of Christiansen, we prove that every mutli-graph $G=(V,E)$ admits a proper edge colouring $\phi:E\to \{1,2,\dots\}$ which is local, that is, $\phi(e)\le \max\{d(x)+\pi(x),d(y)+\pi(y)\}$ for every edge $e$ with end-points $x,y\in V$, where $d(z)$ (resp.\ $\pi(z)$) denotes the degree of a vertex $z$ (resp.\ the maximum edge multiplicity at $z$). This is derived from a local version of the Fan Equation.

##### 4.Strong metric dimension of the prime ideal sum graph of a commutative ring

**Authors:**Praveen Mathil, Jitender Kumar, Reza Nikandish

**Abstract:** Let $R$ be a commutative ring with unity. The prime ideal sum graph of the ring $R$ is the simple undirected graph whose vertex set is the set of all nonzero proper ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if $I + J$ is a prime ideal of $R$. In this paper, we obtain the strong metric dimension of the prime ideal sum graph for various classes of Artinian non-local commutative rings.

##### 5.Enumeration of splitting subsets of endofunctions on finite sets

**Authors:**Divya Aggarwal

**Abstract:** Let $d$ and $n$ be positive integers such that $d|n$. Let $[n]=\{1,2,\ldots,n\}$ and $T$ be an endofunction on $[n]$. A subset $W$ of $[n]$ of cardinality $n/d$ is said to be $d$-splitting if $W \cup TW \cup \cdots \cup T^{d-1}W =[n]$. Let $\sigma(d;T)$ denote the number of $d$-splitting subsets. If $\sigma(2;T)>0$, then we show that $\sigma(2;T)=g_T(-1)$, where $g_T(t)$ is the generating function for the number of $T$-invariant subsets of $[n]$. It is interesting to note that substituting a root of unity into a polynomial with integer coefficients has an enumerative meaning. More generally, let $g_T(t_1,\ldots,t_d)$ be the generating function for the number of $d$-flags of $T$-invariant subsets. We prove for certain endofunctions $T$, if $\sigma(d;T)>0$, then $\sigma(d;T)=g_T(\zeta,\zeta^2,\ldots,\zeta^d)$, where $\zeta$ is a primitive $d^{th}$ root of unity.

##### 6.Integer Carathéodory results with bounded multiplicity

**Authors:**Stefan Kuhlmann

**Abstract:** The integer Carath\'eodory rank of a pointed rational cone $C$ is the smallest number $k$ such that every integer vector contained in $C$ is an integral non-negative combination of at most $k$ Hilbert basis elements. We investigate the integer Carath\'eodory rank of simplicial cones with respect to their multiplicity, i.e., the determinant of the integral generators of the cone. One of the main results states that simplicial cones with multiplicity bounded by five have the integral Carath\'eodory property, that is, the integer Carath\'eodory rank equals the dimension. Furthermore, we present a novel upper bound on the integer Carath\'eodory rank which depends on the dimension and the multiplicity. This bound improves upon the best known upper bound on the integer Carath\'eodory rank if the dimension exceeds the multiplicity. At last, we present special cones which have the integral Carath\'eodory property such as certain dual cones of Gorenstein cones.

##### 7.A note on non-empty cross-intersecting families

**Authors:**Menglong Zhang, Tao Feng

**Abstract:** The families $\mathcal F_1\subseteq \binom{[n]}{k_1},\mathcal F_2\subseteq \binom{[n]}{k_2},\dots,\mathcal F_r\subseteq \binom{[n]}{k_r}$ are said to be cross-intersecting if $|F_i\cap F_j|\geq 1$ for any $1\leq i<j\leq r$ and $F_i\in \mathcal F_i$, $F_j\in\mathcal F_j$. Cross-intersecting families $\mathcal F_1,\mathcal F_2,\dots,\mathcal F_r$ are said to be non-empty if $\mathcal F_i\neq\emptyset$ for any $1\leq i\leq r$. This paper shows that if $\mathcal F_1\subseteq\binom{[n]}{k_1},\mathcal F_2\subseteq\binom{[n]}{k_2},\dots,\mathcal F_r\subseteq\binom{[n]}{k_r}$ are non-empty cross-intersecting families with $k_1\geq k_2\geq\cdots\geq k_r$ and $n\geq k_1+k_2$, then $\sum_{i=1}^{r}|\mathcal F_i|\leq\max\{\binom{n}{k_1}-\binom{n-k_r}{k_1}+\sum_{i=2}^{r}\binom{n-k_r}{k_i-k_r},\ \sum_{i=1}^{r}\binom{n-1}{k_i-1}\}$. This solves a problem posed by Shi, Frankl and Qian recently. The extremal families attaining the upper bounds are also characterized.

##### 8.Equidistribution of set-valued statistics on standard Young tableaux and transversals

**Authors:**Robin D. P. Zhou, Sherry H. F. Yan

**Abstract:** As a natural generalization of permutations, transversals of Young diagrams play an important role in the study of pattern avoiding permutations. Let $\mathcal{T}_{\lambda}(\tau)$ and $\mathcal{ST}_{\lambda}(\tau)$ denote the set of $\tau$-avoiding transversals and $\tau$-avoiding symmetric transversals of a Young diagram $\lambda$, respectively. In this paper, we are mainly concerned with the distribution of the peak set and the valley set on standard Young tableaux and pattern avoiding transversals. In particular, by introducing Knuth transformations on standard Young tableaux, we prove that the peak set and the valley set are equidistributed on the standard Young tableaux of shape $\lambda/\mu$ for any skew diagram $\lambda/\mu$. The equidistribution enables us to show that the peak set is equidistributed over $\mathcal{T}_{\lambda}(12\cdots k\tau)$ (resp. $\mathcal{ST}_{\lambda}(12\cdots k \tau)$) and $\mathcal{T}_{\lambda}(k\cdots 21\tau) $ (resp. $\mathcal{ST}_{\lambda}(k\cdots 21\tau)$) for any Young diagram $\lambda$ and any permutation $\tau$ of $\{k+1, k+2, \ldots, k+m\}$ with $k,m\geq 1$. Our results are refinements of the result of Backelin-West-Xin which states that $|\mathcal{T}_{\lambda}(12\cdots k\tau)|=|\mathcal{T}_{\lambda}(k\cdots 21 \tau)|$ and the result of Bousquet-M\'elou and Steingr\'imsson which states that $|\mathcal{ST}_{\lambda}(12\cdots k \tau)|=|\mathcal{ST}_{\lambda}(k\cdots 21 \tau)|$.

##### 9.On Galois groups of type-1 minimally rigid graphs

**Authors:**Mehdi Makhul, Josef Schicho, Audie Warren

**Abstract:** For every graph that is mimimally rigid in the plane, its Galois group is defined as the Galois group generated by the coordinates of its planar realizations, assuming that the edge lengths are transcendental and algebraically independent. Here we compute the Galois group of all minimally rigid graphs that can be constructed from a single edge by repeated Henneberg 1-steps. It turns out that any such group is totally imprimitive, i.e., it is determined by all the partitions it preserves.

##### 10.A Cheeger inequality for the lower spectral gap

**Authors:**Jyoti Prakash Saha

**Abstract:** Let $\Gamma$ be a Cayley graph, or a Cayley sum graph, or a twisted Cayley graph, or a twisted Cayley sum graph, or a vertex-transitive graph. Denote the degree of $\Gamma$ by $d$ and the edge Cheeger constant of $\Gamma$ by $\mathfrak{h}_\Gamma$. Assume that $\Gamma$ is undirected, non-bipartite and has finitely many vertices. We prove that the edge bipartiteness constant of $\Gamma$ is $\Omega({\mathfrak{h}_\Gamma}/{d})$, and the smallest eigenvalue of the normalized adjacency operator of $\Gamma$ is $-1 + \Omega({\mathfrak{h}_\Gamma^2}/{d^2})$. These answer in the affirmative a question of Moorman, Ralli and Tetali on the edge bipartiteness constant of Cayley graphs, and a question of them on the lower spectral gap of Cayley sum graphs.

##### 11.Unimodality of $k$-Regular Partition into Distinct Parts with Bounded Largest Part

**Authors:**Janet J. W. Dong, Kathy Q. Ji

**Abstract:** A $k$-regular partition into distinct parts is a partition into distinct parts with no part divisible by $k$. In this paper, we provide a general method to establish the unimodality of $k$-regular partition into distinct parts where the largest part is at most $km+k-1$. Let $d_{k,m}(n)$ denote the number of $k$-regular partition of $n$ into distinct parts where the largest part is at most $km+k-1$. In line with this method, we show that $d_{4,m}(n)\geq d_{4,m}(n-1)$ for $m\geq 0$, $1\leq n\leq 3(m+1)^2$ and $n\neq 4$ and $d_{8,m}(n)\geq d_{8,m}(n-1)$ for $m\geq 2$ and $1\leq n\leq 14(m+1)^2$. When $5\leq k\leq 10$ and $k\neq 8$, we show that $d_{k,m}(n)\geq d_{k,m}(n-1)$ for $m\geq 0$ and $1\leq n\leq \left\lfloor\frac{k(k-1)(m+1)^2}{4}\right\rfloor$.

##### 12.On large regular (1,1,k)-mixed graphs

**Authors:**C. Dalfó, G. Erskine, G. Exoo, M. A. Fiol, N. López, A. Messegué, J. Tuite

**Abstract:** An $(r,z,k)$-mixed graph $G$ has every vertex with undirected degree $r$, directed in- and out-degree $z$, and diameter $k$. In this paper, we study the case $r=z=1$, proposing some new constructions of $(1,1,k)$-mixed graphs with a large number of vertices $N$. Our study is based on computer techniques for small values of $k$ and the use of graphs on alphabets for general $k$. In the former case, the constructions are either Cayley or lift graphs. In the latter case, some infinite families of $(1,1,k)$-mixed graphs are proposed with diameter of the order of $2\log_2 N$.

##### 13.$\varepsilon$-Almost collision-flat universal hash functions and mosaics of designs

**Authors:**Moritz Wiese, Holger Boche

**Abstract:** We introduce, motivate and study $\varepsilon$-almost collision-flat (ACFU) universal hash functions $f:\mathcal X\times\mathcal S\to\mathcal A$. Their main property is that the number of collisions in any given value is bounded. Each $\varepsilon$-ACFU hash function is an $\varepsilon$-almost universal (AU) hash function, and every $\varepsilon$-almost strongly universal (ASU) hash function is an $\varepsilon$-ACFU hash function. We study how the size of the seed set $\mathcal S$ depends on $\varepsilon,|\mathcal X|$ and $|\mathcal A|$. Depending on how these parameters are interrelated, seed-minimizing ACFU hash functions are equivalent to mosaics of balanced incomplete block designs (BIBDs) or to duals of mosaics of quasi-symmetric block designs; in a third case, mosaics of transversal designs and nets yield seed-optimal ACFU hash functions, but a full characterization is missing. By either extending $\mathcal S$ or $\mathcal X$, it is possible to obtain an $\varepsilon$-ACFU hash function from an $\varepsilon$-AU hash function or an $\varepsilon$-ASU hash function, generalizing the construction of mosaics of designs from a given resolvable design (Gnilke, Greferath, Pav{\v c}evi\'c, Des. Codes Cryptogr. 86(1)). The concatenation of an ASU and an ACFU hash function again yields an ACFU hash function. Finally, we motivate ACFU hash functions by their applicability in privacy amplification.

##### 1.New scattered sequences of order 3

**Authors:**Daniele Bartoli, Alessandro Giannoni

**Abstract:** Scattered sequences are a generalization of scattered polynomials. So far, only scattered sequences of order one and two have been constructed. In this paper an infine family of scattered sequences of order three is obtained. Equivalence issues are also considered.

##### 2.The maximum sum of sizes of non-empty pairwise cross-intersecting families

**Authors:**Yang Huang, Yuejian Peng

**Abstract:** Two families $\mathcal{A}$ and $\mathcal{B}$ are cross-intersecting if $A\cap B\ne \emptyset$ for any $A\in \mathcal{A}$ and $B\in \mathcal{B}$. We call $t$ families $\mathcal{A}_1, \mathcal{A}_2,\dots, \mathcal{A}_t$ pairwise cross-intersecting families if $\mathcal{A}_i$ and $\mathcal{A}_j$ are cross-intersecting when $1\le i<j \le t$. Additionally, if $\mathcal{A}_j\ne \emptyset$ for each $j\in [t]$, then we say that $\mathcal{A}_1, \mathcal{A}_2,\dots, \mathcal{A}_t$ are non-empty pairwise cross-intersecting. Let $\mathcal{A}_1\subset{[n]\choose k_1}, \mathcal{A}_2\subset{[n]\choose k_2}, \dots, \mathcal{A}_t\subset{[n]\choose k_t}$ be non-empty pairwise cross-intersecting families with $t\geq 2$, $k_1\geq k_2\geq \cdots \geq k_t$, and $n\geq k_1+k_2$, we determine the maximum value of $\sum_{i=1}^t{|\mathcal{A}_i|}$ and characterize all extremal families. This answers a question of Shi, Frankl and Qian [Combinatorica 42 (2022)] and unifies results of Frankl and Tokushige [J. Combin. Theory Ser. A 61 (1992)] and Shi, Frankl and Qian [Combinatorica 42 (2022)]. The key techniques in previous works cannot be extended to our situation. A result of Kruskal-Katona is applied to allow us to consider only families $\mathcal{A}_i$ whose elements are the first $|\mathcal{A}_i|$ elements in lexicographic order. We bound $\sum_{i=1}^t{|\mathcal{A}_i|}$ by a function $f(R)$ of the last element $R$ (in the lexicographic order) of $\mathcal{A}_1$, introduce the concepts `$c$-sequential' and `down-up family', and show that $f(R)$ has several types of local convexities.

##### 3.On Seymour's and Sullivan's Second Neighbourhood Conjectures

**Authors:**Jiangdong Ai, Stefanie Gerke, Gregory Gutin, Shujing Wang, Anders Yeo, Yacong Zhou

**Abstract:** For a vertex $x$ of a digraph, $d^+(x)$ ($d^-(x)$, resp.) is the number of vertices at distance 1 from (to, resp.) $x$ and $d^{++}(x)$ is the number of vertices at distance 2 from $x$. In 1995, Seymour conjectured that for any oriented graph $D$ there exists a vertex $x$ such that $d^+(x)\leq d^{++}(x)$. In 2006, Sullivan conjectured that there exists a vertex $x$ in $D$ such that $d^-(x)\leq d^{++}(x)$. We give a sufficient condition in terms of the number of transitive triangles for an oriented graph to satisfy Sullivan's conjecture. In particular, this implies that Sullivan's conjecture holds for all orientations of planar graphs and of triangle-free graphs. An oriented graph $D$ is an oriented split graph if the vertices of $D$ can be partitioned into vertex sets $X$ and $Y$ such that $X$ is an independent set and $Y$ induces a tournament. We also show that the two conjectures hold for some families of oriented split graphs, in particular, when $Y$ induces a regular or an almost regular tournament.

##### 4.Local Antimagic Coloring of Some Graphs

**Authors:**Ravindra Pawar, Tarkeshwar Singh, Adarsh Handa, Aloysius Godinho

**Abstract:** Given a graph $G =(V,E)$, a bijection $f: E \rightarrow \{1, 2, \dots,|E|\}$ is called a local antimagic labeling of $G$ if the vertex weight $w(u) = \sum_{uv \in E} f(uv)$ is distinct for all adjacent vertices. The vertex weights under the local antimagic labeling of $G$ induce a proper vertex coloring of a graph $G$. The \textit{local antimagic chromatic number} of $G$ denoted by $\chi_{la}(G)$ is the minimum number of weights taken over all such local antimagic labelings of $G$. In this paper, we investigate the local antimagic chromatic numbers of the union of some families of graphs, corona product of graphs, and necklace graph and we construct infinitely many graphs satisfying $\chi_{la}(G) = \chi(G)$.

##### 5.Transversals via regularity

**Authors:**Yangyang Cheng, Katherine Staden

**Abstract:** Given graphs $G_1,\ldots,G_s$ all on the same vertex set and a graph $H$ with $e(H) \leq s$, a copy of $H$ is transversal or rainbow if it contains at most one edge from each $G_c$. When $s=e(H)$, such a copy contains exactly one edge from each $G_i$. We study the case when $H$ is spanning and explore how the regularity blow-up method, that has been so successful in the uncoloured setting, can be used to find transversals. We provide the analogues of the tools required to apply this method in the transversal setting. Our main result is a blow-up lemma for transversals that applies to separable bounded degree graphs $H$. Our proofs use weak regularity in the $3$-uniform hypergraph whose edges are those $xyc$ where $xy$ is an edge in the graph $G_c$. We apply our lemma to give a large class of spanning $3$-uniform linear hypergraphs $H$ such that any sufficiently large uniformly dense $n$-vertex $3$-uniform hypergraph with minimum vertex degree $\Omega(n^2)$ contains $H$ as a subhypergraph. This extends work of Lenz, Mubayi and Mycroft.

##### 6.From coordinate subspaces over finite fields to ideal multipartite uniform clutters

**Authors:**Ahmad Abdi, Dabeen Lee

**Abstract:** Take a prime power $q$, an integer $n\geq 2$, and a coordinate subspace $S\subseteq GF(q)^n$ over the Galois field $GF(q)$. One can associate with $S$ an $n$-partite $n$-uniform clutter $\mathcal{C}$, where every part has size $q$ and there is a bijection between the vectors in $S$ and the members of $\mathcal{C}$. In this paper, we determine when the clutter $\mathcal{C}$ is ideal, a property developed in connection to Packing and Covering problems in the areas of Integer Programming and Combinatorial Optimization. Interestingly, the characterization differs depending on whether $q$ is $2,4$, a higher power of $2$, or otherwise. Each characterization uses crucially that idealness is a minor-closed property: first the list of excluded minors is identified, and only then is the global structure determined. A key insight is that idealness of $\mathcal{C}$ depends solely on the underlying matroid of $S$. Our theorems also extend from idealness to the stronger max-flow min-cut property. As a consequence, we prove the Replication and $\tau=2$ Conjectures for this class of clutters.

##### 7.Pathwidth vs cocircumference

**Authors:**Marcin Briański, Gwenaël Joret, Michał T. Seweryn

**Abstract:** The {\em circumference} of a graph $G$ with at least one cycle is the length of a longest cycle in $G$. A classic result of Birmel\'e (2003) states that the treewidth of $G$ is at most its circumference minus $1$. In case $G$ is $2$-connected, this upper bound also holds for the pathwidth of $G$; in fact, even the treedepth of $G$ is upper bounded by its circumference (Bria\'nski, Joret, Majewski, Micek, Seweryn, Sharma; 2023). In this paper, we study whether similar bounds hold when replacing the circumference of $G$ by its {\em cocircumference}, defined as the largest size of a {\em bond} in $G$, an inclusion-wise minimal set of edges $F$ such that $G-F$ has more components than $G$. In matroidal terms, the cocircumference of $G$ is the circumference of the bond matroid of $G$. Our first result is the following `dual' version of Birmel\'e's theorem: The treewidth of a graph $G$ is at most its cocircumference. Our second and main result is an upper bound of $3k-2$ on the pathwidth of a $2$-connected graph $G$ with cocircumference $k$. Contrary to circumference, no such bound holds for the treedepth of $G$. Our two upper bounds are best possible up to a constant factor.

##### 8.Permutations that separate close elements, and rectangle packings in the torus

**Authors:**Simon R. Blackburn, Tuvi Etzion

**Abstract:** Let $n$, $s$ and $k$ be positive integers. For distinct $i,j\in\mathbb{Z}_n$, define $||i,j||_n$ to be the distance between $i$ and $j$ when the elements of $\mathbb{Z}_n$ are written in a circle. So \[ ||i,j||_n=\min\{(i-j)\bmod n,(j-i)\bmod n\}. \] A permutation $\pi:\mathbb{Z}_n\rightarrow\mathbb {Z}_n$ is \emph{$(s,k)$-clash-free} if $||\pi(i),\pi(j)||_n\geq k$ whenever $||i,j||_n<s$. So an $(s,k)$-clash-free permutation moves every pair of close elements (at distance less than $s$) to a pair of elements at large distance (at distance at least $k$). The notion of an $(s,k)$-clash-free permutation can be reformulated in terms of certain packings of $s\times k$ rectangles on an $n\times n$ torus. For integers $n$ and $k$ with $1\leq k<n$, let $\sigma(n,k)$ be the largest value of $s$ such that an $(s,k)$-clash-free permutation of $\mathbb{Z}_n$ exists. Strengthening a recent paper of Blackburn, which proved a conjecture of Mammoliti and Simpson, we determine the value of $\sigma(n,k)$ in all cases.

##### 9.On the Split Reliability of Graphs

**Authors:**Jason I. Brown, Isaac McMullin

**Abstract:** A common model of robustness of a graph against random failures has all vertices operational, but the edges independently operational with probability $p$. One can ask for the probability that all vertices can communicate ({\em all-terminal reliability}) or that two specific vertices (or {\em terminals}) can communicate with each other ({\em two-terminal reliability}). A relatively new measure is {\em split reliability}, where for two fixed vertices $s$ and $t$, we consider the probability that every vertex communicates with one of $s$ or $t$, but not both. In this paper, we explore the existence for fixed numbers $n \geq 2$ and $m \geq n-1$ of an {\em optimal} connected $(n,m)$-graph $G_{n,m}$ for split reliability, that is, a connected graph with $n$ vertices and $m$ edges for which for any other such graph $H$, the split reliability of $G_{n,m}$ is at least as large as that of $H$, for {\em all} values of $p \in [0,1]$. Unlike the similar problems for all-terminal and two-terminal reliability, where only partial results are known, we completely solve the issue for split reliability, where we show that there is an optimal $(n,m)$-graph for split reliability if and only if $n\leq 3$, $m=n-1$, or $n=m=4$.

##### 1.On some conjectural series containing binomial coefficients and harmonic numbers

**Authors:**Chuanan Wei

**Abstract:** Binomial coefficients and harmonic numbers are important in many branches of number theory. With the help of the operator method and several summation and transformation formulas for hypergeometric series, we prove eight conjectural series of Z.-W. Sun containing binomial coefficients and harmonic numbers in this paper.

##### 2.Alternating Parity Weak Sequencing

**Authors:**Simone Costa, Stefano Della Fiore

**Abstract:** A subset $S$ of a group $G$ is $t$-weakly sequenceable if there is an ordering $(a_1, \ldots, a_k)$ of its elements such that the partial sums $(s_0, s_1, \ldots, s_k)$, given by $s_0 = 0$ and $s_i = \sum_{j=1}^i a_j$ for $1 \leq i \leq k$, are different whenever $i$ and $j$ are distinct and $|i-j|\leq t$. In [10] it was proved that if the order of a group is $pe$ then all sufficiently large subsets of the non-identity elements are $t$-weakly sequenceable when $p > 3$ is prime, $e \leq 3$ and $t \leq 6$. Inspired by this result, we show that, if $G$ is of the type $G = \mathbb{Z}_p \rtimes_{\varphi} \mathbb{Z}_2$ and the set $S$ is balanced (i.e. contains the same number of even elements which are those in the coset $\mathbb{Z}_p\rtimes_{\varphi} \{0\}$ and odd ones that are those in its complement) then $S$ admits, regardless of its size, an alternating parity $t$-weak sequencing whenever $p > 3$ is prime and $t \leq 8$. On the other hand, we have been able to prove also the following asymptotic result. Let us consider groups of type $G = H \rtimes_{\varphi} \mathbb{Z}_2$, then all sufficiently large balanced subsets of the non-identity elements admit an alternating parity $t$-weak sequencing. This result has been obtained using a hybrid approach that combines both Ramsey theory and the probabilistic method. The same procedure works also for studying the weak sequenceability for generic sufficiently large (not necessarily balanced) sets. Here we have been able to prove that, if the size of a subset $S$ of a group $G$ is large enough and if $S$ does not contain $0$, then $S$ is $t$-weakly sequenceable.

##### 3.Realizable Dimension of Periodic Frameworks

**Authors:**Ryoshun Oba, Shin-ichi Tanigawa

**Abstract:** Belk and Connelly introduced the realizable dimension $\textrm{rd}(G)$ of a finite graph $G$, which is the minimum nonnegative integer $d$ such that every framework $(G,p)$ in any dimension admits a framework in $\mathbb{R}^d$ with the same edge lengths. They characterized finite graphs with realizable dimension at most $1$, $2$, or $3$ in terms of forbidden minors. In this paper, we consider periodic frameworks and extend the notion to $\mathbb{Z}$-symmetric graphs. We give a forbidden minor characterization of $\mathbb{Z}$-symmetric graphs with realizable dimension at most $1$ or $2$, and show that the characterization can be checked in polynomial time for given quotient $\mathbb{Z}$-labelled graphs.

##### 4.On bicyclic graphs with maximum edge Mostar index

**Authors:**Fazal Hayat, Shou-Jun Xu, Bo Zhou

**Abstract:** For a given connected graph $G$, the edge Mostar index $Mo_e(G)$ is defined as $Mo_e(G)=\sum_{e=uv \in E(G)}|m_u(e|G) - m_v(e|G)|$, where $m_u(e|G)$ and $m_v(e|G)$ are respectively, the number of edges of $G$ lying closer to vertex $u$ than to vertex $v$ and the number of edges of $G$ lying closer to vertex $v$ than to vertex $u$. In this paper, we determine sharp upper bound for edge Mostar index on bicyclic graphs with fixed number of edges, also the graphs that achieve the bound are completely characterized, and thus disprove a conjecture on the edge Mostar index of bicyclic graph in [H. Liu, L. Song, Q. Xiao, Z. Tang, On edge Mostar index of graphs. Iranian J. Math. Chem. 11(2) (2020) 95--106].

##### 1.New bounds for odd colourings of graphs

**Authors:**Tianjiao Dai, Qiancheng Ouyang, François Pirot

**Abstract:** Given a graph $G$, a vertex-colouring $\sigma$ of $G$, and a subset $X\subseteq V(G)$, a colour $x \in \sigma(X)$ is said to be \emph{odd} for $X$ in $\sigma$ if it has an odd number of occurrences in $X$. We say that $\sigma$ is an \emph{odd colouring} of $G$ if it is proper and every (open) neighbourhood has an odd colour in $\sigma$. The odd chromatic number of a graph $G$, denoted by $\chi_o(G)$, is the minimum $k\in\mathbb{N}$ such that an odd colouring $\sigma \colon V(G)\to [k]$ exists. In a recent paper, Caro, Petru\v sevski and \v Skrekovski conjectured that every connected graph of maximum degree $\Delta\ge 3$ has odd-chromatic number at most $\Delta+1$. We prove that this conjecture holds asymptotically: for every connected graph $G$ with maximum degree $\Delta$, $\chi_o(G)\le\Delta+O(\ln\Delta)$ as $\Delta \to \infty$. We also prove that $\chi_o(G)\le\lfloor3\Delta/2\rfloor+2$ for every $\Delta$. If moreover the minimum degree $\delta$ of $G$ is sufficiently large, we have $\chi_o(G) \le \chi(G) + O(\Delta \ln \Delta/\delta)$ and $\chi_o(G) = O(\chi(G)\ln \Delta)$. Finally, given an integer $h\ge 1$, we study the generalisation of these results to $h$-odd colourings, where every vertex $v$ must have at least $\min \{\deg(v),h\}$ odd colours in its neighbourhood. Many of our results are tight up to some multiplicative constant.

##### 2.Injective coloring of product graphs

**Authors:**Babak Samadi, Nasrin Soltankhah, Ismael G. Yero

**Abstract:** The problem of injective coloring in graphs can be revisited through two different approaches: coloring the two-step graphs and vertex partitioning of graphs into open packing sets, each of which is equivalent to the injective coloring problem itself. Taking these facts into account, we observe that the injective coloring lies between graph coloring and domination theory. We make use of these three points of view in this paper so as to investigate the injective coloring of some well-known graph products. We bound the injective chromatic number of direct and lexicographic product graphs from below and above. In particular, we completely determine this parameter for the direct product of two cycles. We also give a closed formula for the corona product of two graphs.

##### 3.On Fan's conjecture about $4$-flow

**Authors:**Deping Song, Shuang Li, Xiao Wang

**Abstract:** Let $G$ be a bridgeless graph, $C$ is a circuit of $G$. Fan proposed a conjecture that if $G/C$ admits a nowhere-zero 4-flow, then $G$ admits a 4-flow $(D,f)$ such that $E(G)-E(C)\subseteq$ supp$(f)$ and $|\textrm{supp}(f)\cap E(C)|>\frac{3}{4}|E(C)|$. The purpose of this conjecture is to find shorter circuit cover in bridgeless graphs. Fan showed that the conjecture holds for $|E(C)|\le19.$ Wang, Lu and Zhang showed that the conjecture holds for $|E(C)|\le 27$. In this paper, we prove that the conjecture holds for $|E(C)|\le 35.$

##### 4.Using alternating de Bruijn sequences to construct de Bruijn tori

**Authors:**Matthew Kreitzer, Mihai Nica, Rajesh Pereira

**Abstract:** A de Bruijn torus is the two dimensional generalization of a de Bruijn sequence. While some methods exist to generate these tori, only a few methods of construction are known. We present a novel method to generate de Bruijn tori with rectangular windows by combining two variants de Bruijn sequences called `Alternating de Bruijn sequences' and `De Bruijn families'.

##### 5.Strong domination number of graphs from primary subgraphs

**Authors:**Saeid Alikhani, Nima Ghanbari, Michael A. Henning

**Abstract:** A set $D$ of vertices is a strong dominating set in a graph $G$, if for every vertex $x\in V(G) \setminus D$ there is a vertex $y\in D$ with $xy\in E(G)$ and $deg(x) \leq deg(y)$. The strong domination number $\gamma_{st}(G)$ of $G$ is the minimum cardinality of a strong dominating set in $G$. Let $G$ be a connected graph constructed from pairwise disjoint connected graphs $G_1,\ldots ,G_k$ by selecting a vertex of $G_1$, a vertex of $G_2$, and identifying these two vertices, and thereafter continuing in this manner inductively. The graphs $G_1,\ldots ,G_k$ are the primary subgraphs of $G$. In this paper, we study the strong domination number of $K_r$-gluing of two graphs and investigate the strong domination number for some particular cases of graphs from their primary subgraphs.

##### 6.Stability of Rose Window graphs

**Authors:**Milad Ahanjideh, István Kovács, Klavdija Kutnar

**Abstract:** A graph $\Gamma$ is said to be stable if for the direct product $\Gamma\times\mathbf{K}_2$, ${\rm Aut}(\Gamma \times \mathbf{K}_2)$ is isomorphic to ${\rm Aut}(\Gamma) \times \mathbb{Z}_2$; otherwise, it is called unstable. An unstable graph is called non-trivially unstable when it is not bipartite and no two vertices have the same neighborhood. Wilson described nine families of unstable Rose Window graphs and conjectured that these contain all non-trivially unstable Rose Window graphs (2008). In this paper we show that the conjecture is true.

##### 7.Distinct eigenvalues of the Transposition graph

**Authors:**Elena V. Konstantinova, Artem Kravchuk

**Abstract:** Transposition graph $T_n$ is defined as a Cayley graph over the symmetric group generated by all transpositions. It is known that all eigenvalues of $T_n$ are integers. Moreover, zero is its eigenvalue for any $n\geqslant 4$. But the exact distribution of the spectrum of the graph $T_n$ is unknown. In this paper we prove that integers from the interval $[-\frac{n-4}{2}, \frac{n-4}{2}]$ lie in the spectrum of $T_n$ if $n \geqslant 19$.

##### 8.Excluding Surfaces as Minors in Graphs

**Authors:**Dimitrios M. Thilikos, Sebastian Wiederrecht

**Abstract:** We introduce an annotated extension of treewidth that measures the contribution of a vertex set $X$ to the treewidth of a graph $G.$ This notion provides a graph distance measure to some graph property $\mathcal{P}$: A vertex set $X$ is a $k$-treewidth modulator of $G$ to $\mathcal{P}$ if the treewidth of $X$ in $G$ is at most $k$ and its removal gives a graph in $\mathcal{P}.$This notion allows for a version of the Graph Minors Structure Theorem (GMST) that has no need for apices and vortices: $K_k$-minor free graphs are those that admit tree-decompositions whose torsos have $c_{k}$-treewidth modulators to some surface of Euler-genus $c_{k}.$ This reveals that minor-exclusion is essentially tree-decomposability to a ``modulator-target scheme'' where the modulator is measured by its treewidth and the target is surface embeddability. We then fix the target condition by demanding that $\Sigma$ is some particular surface and define a ``surface extension'' of treewidth, where $\Sigma\mbox{-}\mathsf{tw}(G)$ is the minimum $k$ for which $G$ admits a tree-decomposition whose torsos have a $k$-treewidth modulator to being embeddable in $\Sigma.$We identify a finite collection $\mathfrak{D}_{\Sigma}$ of parametric graphs and prove that the minor-exclusion of the graphs in $\mathfrak{D}_{\Sigma}$ precisely determines the asymptotic behavior of ${\Sigma}\mbox{-}\mathsf{tw},$ for every surface $\Sigma.$ It follows that the collection $\mathfrak{D}_{\Sigma}$ bijectively corresponds to the ``surface obstructions'' for $\Sigma,$ i.e., surfaces that are minimally non-contained in $\Sigma.$

##### 9.The diameter of randomly twisted hypercubes

**Authors:**Lucas Aragão, Maurício Collares, Gabriel Dahia, João Pedro Marciano

**Abstract:** The $n$-dimensional random twisted hypercube $\mathbf{G}_n$ is constructed recursively by taking two instances of $\mathbf{G}_{n-1}$, with any joint distribution, and adding a random perfect matching between their vertex sets. Benjamini, Dikstein, Gross, and Zhukovskii showed that its diameter is $O(n\log \log \log n/\log \log n)$ with high probability and at least ${(n - 1)/ \log_2 n}$. We improve their upper bound by showing that $$\operatorname{diam}(\mathbf{G}_n) = \big(1 + o(1)\big) \frac{n}{\log_2 n}$$ with high probability.

##### 1.A generalization of diversity for intersecting families

**Authors:**Van Magnan, Cory Palmer, Ryan Wood

**Abstract:** Let $\mathcal{F}\subseteq \binom{[n]}{r}$ be an intersecting family of sets and let $\Delta(\mathcal{F})$ be the maximum degree in $\mathcal{F}$, i.e., the maximum number of edges of $\mathcal{F}$ containing a fixed vertex. The \emph{diversity} of $\mathcal{F}$ is defined as $d(\mathcal{F}) := |\mathcal{F}| - \Delta(\mathcal{F})$. Diversity can be viewed as a measure of distance from the `trivial' maximum-size intersecting family given by the Erd\H os-Ko-Rado Theorem. Indeed, the diversity of this family is $0$. Moreover, the diversity of the largest non-trivial intersecting family \`a la Hilton-Milner is $1$. It is known that the maximum possible diversity of an intersecting family $\mathcal{F}\subseteq \binom{[n]}{r}$ is $\binom{n-3}{r-2}$ as long as $n$ is large enough. We introduce a generalization called the \emph{$C$-weighted diversity} of $\mathcal{F}$ as $d_C(\mathcal{F}) := |\mathcal{F}| - C \cdot \Delta(\mathcal{F})$. We determine the maximum value of $d_C(\mathcal{F})$ for intersecting families $\mathcal{F} \subseteq \binom{[n]}{r}$ and characterize the maximal families for $C\in \left[0,\frac{7}{3}\right)$ as well as give general bounds for all $C$. Our results imply, for large $n$, a recent conjecture of Frankl and Wang concerning a related diversity-like measure. Our primary technique is a variant of Frankl's Delta-system method.

##### 2.Extremal Peisert-type graphs without the strict-EKR property

**Authors:**Sergey Goryainov, Chi Hoi Yip

**Abstract:** Let $q$ be a prime power. We study extremal Peisert-type graphs of order $q^2$ without the strict-EKR property, that is, Peisert-type graphs of order $q^2$ without the strict-EKR property and with the minimum number of edges. First, we determine this minimum number of edges for each value of $q$. If $q$ is a square, we show the uniqueness of extremal graph and its isomorphism with certain affine polar graph. Using the isomorphism, we conclude that there is no Hilton-Milner type result for this extremal graph. We also prove the tightness of the weight-distribution bound for both non-principal eigenvalues of this graph. If $q$ is a cube but not a square, we show the uniqueness of extremal graph and determine the number and the structure of non-canonical cliques. Finally, we show such uniqueness result does not extend to all $q$.

##### 3.New bijective proofs pertaining to alternating sign matrices

**Authors:**Takuya Inoue

**Abstract:** The alternating sign matrices-descending plane partitions (ASM-DPP) bijection problem is one of the most intriguing open problems in bijective combinatorics, which is also relevant to integrable combinatorics. The notion of a signed set and a signed bijection is used in [Fischer, I. \& Konvalinka, M., Electron. J. Comb., 27 (2020) 3-35.] to construct a bijection between $\text{ASM}_n \times \text{DPP}_{n-1}$ and $\text{DPP}_n \times \text{ASM}_{n-1}$. Here, we shall construct a more natural alternative to a signed bijection between alternating sign matrices and shifted Gelfand-Tsetlin patterns which is presented in that paper, based on the notion of compatibility which we introduce to measure the naturalness of a signed bijection. In addition, we give a bijective proof for the refined enumeration of an extension of alternating sign matrices with $n+3$ statistics, first proved in [Fischer, I. \& Schreier-Aigner, F., Advances in Mathematics 413 (2023) 108831.].

##### 4.On $k$-neighborly reorientations of oriented matroids

**Authors:**Rangel Hernández-Ortiz, Kolja Knauer, Luis Pedro Montejano

**Abstract:** We study the existence and the number of $k$-neighborly reorientations of an oriented matroid. This leads to $k$-variants of McMullen's problem and Roudneff's conjecture, the case $k=1$ being the original statements on complete cells in arrangements. Adding to results of Larman and Garc\'ia-Col\'in, we provide new bounds on the $k$-McMullen's problem and prove the conjecture for several ranks and $k$ by computer. Further, we show that $k$-Roudneff's conjecture for fixed rank and $k$ reduces to a finite case analyse. As a consequence we prove the conjecture for odd rank $r$ and $k=\frac{r-1}{2}$ as well as for rank $6$ and $k=2$ with the aid of the computer.

##### 5.Combinatorial commutative algebra rules

**Authors:**Ada Stelzer, Alexander Yong

**Abstract:** An algorithm is presented that generates sets of size equal to the degree of a given variety defined by a homogeneous ideal. This algorithm suggests a versatile framework to study various problems in combinatorial algebraic geometry and related fields.

##### 6.Low-complexity approximations for sets defined by generalizations of affine conditions

**Authors:**W. T. Gowers, Thomas Karam

**Abstract:** Let $p$ be a prime, let $S$ be a non-empty subset of $\mathbb{F}_p$ and let $0<\epsilon\leq 1$. We show that there exists a constant $C=C(p, \epsilon)$ such that for every positive integer $k$, whenever $\phi_1, \dots, \phi_k: \mathbb{F}_p^n \rightarrow \mathbb{F}_p$ are linear forms and $E_1, \dots, E_k$ are subsets of $\mathbb{F}_p$, there exist linear forms $\psi_1, \dots, \psi_C: \mathbb{F}_p^n \rightarrow \mathbb{F}_p$ and subsets $F_1, \dots, F_C$ of $\mathbb{F}_p$ such that the set $U=\{x \in S^n: \psi_1(x) \in F_1, \dots, \psi_C(x) \in F_C\}$ is contained inside the set $V=\{x \in S^n: \phi_1(x) \in E_1, \dots, \phi_k(x) \in E_k\}$, and the difference $V \setminus U$ has density at most $\epsilon$ inside $S^n$. We then generalize this result to one where $\phi_1, \dots, \phi_k$ are replaced by homomorphisms $G^n \to H$ for some pair of finite Abelian groups $G$ and $H$, and to another where they are replaced by polynomial maps $\mathbb{F}_p^n \to \mathbb{F}_p$ of small degree.

##### 7.Separating path systems in trees

**Authors:**Francisco Arrepol, Patricio Asenjo, Raúl Astete, Víctor Cartes, Anahí Gajardo, Valeria Henríquez, Catalina Opazo, Nicolás Sanhueza-Matamala, Christopher Thraves Caro

**Abstract:** For a graph $G$, an edge-separating (resp. vertex-separating) path system of $G$ is a family of paths in $G$ such that for any pair of edges $e_1, e_2$ (resp. pair of vertices $v_1, v_2$) of $G$ there is at least one path in the family that contains one of $e_1$ and $e_2$ (resp. $v_1$ and $v_2$) but not the other. We determine the size of a minimum edge-separating path system of an arbitrary tree $T$ as a function of its number of leaves and degree-two vertices. We obtain bounds for the size of a minimal vertex-separating path system for trees, which we show to be tight in many cases. We obtain similar results for a variation of the definition, where we require the path system to separate edges and vertices simultaneously. Finally, we investigate the size of a minimal vertex-separating path system in Erd\H{o}s--R\'enyi random graphs.

##### 8.Fusions of the Tensor Square of a Strongly Regular Graph

**Authors:**Allen Herman, Neha Joshi

**Abstract:** In this paper we determine all fusions of the association scheme $\mathcal{A} \otimes \mathcal{A}$, where $\mathcal{A}$ is the symmetric rank $3$ association scheme corresponding to a strongly regular graph. This includes both guaranteed fusions, which are fusions for all symmetric rank $3$ association schemes $\mathcal{A}$, and specific case fusions, which only exist under restrictions on the parameters of the association scheme. Along the way we will determine the fusions of wreath products of strongly regular graphs and the fusions of the tensor square of a symmetric rank $3$ table algebra. This extends recent work of the authors and Meagher, which solved the same problem for the generalized Hamming scheme $H(2,\mathcal{A})$ of the association scheme obtained from a strongly regular graph. The main results of this article show (1) the families of strongly regular graphs for which $\mathcal{A} \otimes \mathcal{A}$ has a special case fusion are the same families for which $H(2,\mathcal{A})$ has a special case fusion; and (2) the imprimitive strongly regular graphs are the only family of strongly regular graphs for which the wreath product $\mathcal{A} \wr \mathcal{A}$ has a special case fusion.

##### 1.Lattice paths in Young diagrams

**Authors:**Thomas K. Waring

**Abstract:** Fill each box in a Young diagram with the number of paths from the bottom of its column to the end of its row, using steps north and east. Then, any square sub-matrix of this array starting on the south-east boundary has determinant one. We provide a - to our knowledge - new bijective argument for this result. Using the same ideas, we prove further identities involving these numbers which correspond to an integral orthonormal basis of the inner product space with Gram matrix given by the array in question. This provides an explicit answer to a question (listed as unsolved) raised in Exercise 6.27 c) of Stanley's Enumerative Combinatorics.

##### 2.Graphs whose mixed metric dimension is equal to their order

**Authors:**Ali Ghalavand, Sandi Klavžar, Mostafa Tavakoli

**Abstract:** The mixed metric dimension ${\rm mdim}(G)$ of a graph $G$ is the cardinality of a smallest set of vertices that (metrically) resolves each pair of elements from $V(G)\cup E(G)$. We say that $G$ is a max-mdim graph if ${\rm mdim}(G) = n(G)$. It is proved that a max-mdim graph $G$ with $n(G)\ge 7$ contains a vertex of degree at least $5$. Using the strong product of graphs and amalgamations large families of max-mdim graphs are constructed. The mixed metric dimension of graphs with at least one universal vertex is determined. The mixed metric dimension of graphs $G$ with cut vertices is bounded from the above and the mixed metric dimension of block graphs computed.

##### 3.Cubic factor-invariant graphs of cycle quotient type -- the alternating case

**Authors:**Brian Alspach, Primoz Sparl

**Abstract:** We investigate connected cubic vertex-transitive graphs whose edge sets admit a partition into a $2$-factor $\mathcal{C}$ and a $1$-factor that is invariant under a vertex-transitive subgroup of the automorphism group of the graph and where the quotient graph with respect to $\mathcal{C}$ is a cycle. There are two essentially different types of such cubic graphs. In this paper we focus on the examples of what we call the alternating type. We classify all such examples admitting a vertex-transitive subgroup of the automorphism group of the graph preserving the corresponding $2$-factor and also determine the ones for which the $2$-factor is invariant under the full automorphism group of the graph. In this way we introduce a new infinite family of cubic vertex-transitive graphs that is a natural generalization of the well-known generalized Petersen graphs as well as of the honeycomb toroidal graphs. The family contains an infinite subfamily of arc-regular examples and an infinite family of $2$-arc-regular examples.

##### 4.Partial domination in supercubic graphs

**Authors:**Csilla Bujtás andMichael A. Henning, Sandi Klavžar

**Abstract:** For some $\alpha$ with $0 < \alpha \le 1$, a subset $X$ of vertices in a graph $G$ of order~$n$ is an $\alpha$-partial dominating set of $G$ if the set $X$ dominates at least $\alpha \times n$ vertices in $G$. The $\alpha$-partial domination number ${\rm pd}_{\alpha}(G)$ of $G$ is the minimum cardinality of an $\alpha$-partial dominating set of $G$. In this paper partial domination of graphs with minimum degree at least $3$ is studied. It is proved that if $G$ is a graph of order~$n$ and with $\delta(G)\ge 3$, then ${\rm pd}_{\frac{7}{8}}(G) \le \frac{1}{3}n$. If in addition $n\ge 60$, then ${\rm pd}_{\frac{9}{10}}(G) \le \frac{1}{3}n$, and if $G$ is a connected cubic graph of order $n\ge 28$, then ${\rm pd}_{\frac{13}{14}}(G) \le \frac{1}{3}n$. Along the way it is shown that there are exactly four connected cubic graphs of order $14$ with domination number $5$.

##### 5.A family of Counterexamples on Inequality among Symmetric Functions

**Authors:**Jia Xu, Yong Yao

**Abstract:** Inequalities among symmetric functions are fundamental questions in mathematics and have various applications in science and engineering. In this paper, we tackle a conjecture about inequalities among the complete homogeneous symmetric function $H_{n,\lambda}$, that is, the inequality $H_{n,\lambda}\leq H_{n,\mu}$ implies majorization order $\lambda\preceq\mu$. This conjecture was proposed by Cuttler, Greene and Skandera in 2011. The conjecture is a close analogy with other known results on Muirhead-type inequalities. In 2021, Heaton and Shankar disproved the conjecture by showing a counterexample for degree $d=8$ and number of variables $n=3$. They then asked whether the conjecture is true when~ the number of variables, $n$, is large enough? In this paper, we answer the question by proving that the conjecture does not hold when $d\geq8$ and $n\geq2$. A crucial step of the proof relies on variables reduction. Inspired by this, we propose a new conjecture for $H_{n,\lambda}\leq H_{n,\mu}$.

##### 6.On Newton's identities in positive characteristic

**Authors:**Sjoerd de Vries

**Abstract:** Newton's identities provide a way to express elementary symmetric polynomials in terms of power polynomials over fields of characteristic zero. In this article we study symmetric polynomials in positive characteristic. Our main result shows that in this setting, one can recover the elementary symmetric polynomials as rational functions in the power polynomials.

##### 7.Characterization of flip process rules with the same trajectories

**Authors:**Eng Keat Hng

**Abstract:** Garbe, Hladk\'y, \v{S}ileikis and Skerman recently introduced a general class of random graph processes called flip processes and proved that the typical evolution of these discrete-time random graph processes correspond to certain continuous-time deterministic graphon trajectories. We obtain a complete characterization of the equivalence classes of flip process rules with the same graphon trajectories. As an application, we characterize the flip process rules which are unique in their equivalence classes. These include several natural families of rules such as the complementing rules, the component completion rules, the extremist rules, and the clique removal rules.

##### 8.Turán problems for oriented graphs

**Authors:**Andrzej Grzesik, Justyna Jaworska, Bartłomiej Kielak, Aliaksandra Novik, Tomasz Ślusarczyk

**Abstract:** A classical Tur\'an problem asks for the maximum possible number of edges in a graph of a given order that does not contain a particular graph $H$ as a subgraph. It is well-known that the chromatic number of $H$ is the graph parameter which describes the asymptotic behavior of this maximum. Here, we consider an analogous problem for oriented graphs, where compressibility plays the role of the chromatic number. Since any oriented graph having a directed cycle is not contained in any transitive tournament, it makes sense to consider only acyclic oriented graphs as forbidden subgraphs. We provide basic properties of the compressibility, show that the compressibility of acyclic oriented graphs with out-degree at most 2 is polynomial with respect to the maximum length of a directed path, and that the same holds for a larger out-degree bound if the Erd\H{o}s-Hajnal conjecture is true. Additionally, generalizing previous results on powers of paths and arbitrary orientations of cycles, we determine the compressibility of acyclic oriented graphs with a restricted structure.

##### 9.Permutoric Promotion: Gliding Globs, Sliding Stones, and Colliding Coins

**Authors:**Colin Defant, Rachana Madhukara, Hugh Thomas

**Abstract:** The first author recently introduced toric promotion, an operator that acts on the labelings of a graph $G$ and serves as a cyclic analogue of Sch\"utzenberger's promotion operator. Toric promotion is defined as the composition of certain toggle operators, listed in a natural cyclic order. We consider more general permutoric promotion operators, which are defined as compositions of the same toggles, but in permuted orders. We settle a conjecture of the first author by determining the orders of all permutoric promotion operators when $G$ is a path graph. In fact, we completely characterize the orbit structures of these operators, showing that they satisfy the cyclic sieving phenomenon. The first half of our proof requires us to introduce and analyze new broken promotion operators, which can be interpreted via globs of liquid gliding on a path graph. For the latter half of our proof, we reformulate the dynamics of permutoric promotion via stones sliding along a cycle graph and coins colliding with each other on a path graph.

##### 10.The list-Ramsey threshold for families of graphs

**Authors:**Eden Kuperwasser, Wojciech Samotij

**Abstract:** Given a family of graphs $\mathcal{F}$ and an integer $r$, we say that a graph is $r$-Ramsey for $\mathcal{F}$ if any $r$-colouring of its edges admits a monochromatic copy of a graph from $\mathcal{F}$. The threshold for the classic Ramsey property, where $\mathcal{F}$ consists of one graph, was located in the celebrated work of R\"odl and Ruci\'nski. In this paper, we offer a twofold generalisation to the R\"odl--Ruci\'nski theorem. First, we show that the list-colouring version of the property has the same threshold. Second, we extend this result to finite families $\mathcal{F}$, where the threshold statements might also diverge. This also confirms further special cases of the Kohayakawa--Kreuter conjecture. Along the way, we supply a short(-ish), self-contained proof of the $0$-statement of the R\"odl--Ruci\'nski theorem.

##### 11.On the faces of unigraphic $3$-polytopes

**Authors:**Riccardo W. Maffucci

**Abstract:** A $3$-polytope is a $3$-connected, planar graph. It is called unigraphic if it does not share its vertex degree sequence with any other $3$-polytope, up to graph isomorphism. The classification of unigraphic $3$-polytopes appears to be a difficult problem. In this paper we prove that, apart from pyramids, all unigraphic $3$-polytopes have no $n$-gonal faces for $n\geq 8$. Our method involves defining several planar graph transformations on a given $3$-polytope containing an $n$-gonal face with $n\geq 8$. The delicate part is to prove that, for every such $3$-polytope, at least one of these transformations both preserves $3$-connectivity, and is not an isomorphism.

##### 1.Weakly reflecting graph properties

**Authors:**Attila Joó

**Abstract:** L. Soukup formulated an abstract framework in his introductory paper for proving theorems about uncountable graphs by subdividing them by an increasing continuous chain of elementary submodels. The applicability of this method relies on the preservation of a certain property (that varies from problem to problem) by the subgraphs obtained by subdividing the graph by an elementary submodel. He calls the properties that are preserved ``well-reflecting''. The aim of this paper is to investigate the possibility of weakening of the assumption ``well-reflecting'' in L. Soukup's framework. Our motivation is to gain better understanding about a class of problems in infinite graph theory where a weaker form of well-reflection naturally occurs.

##### 2.Small codes

**Authors:**Igor Balla

**Abstract:** In 1930, Tammes posed the problem of determining $\rho(r, n)$, the minimum over all sets of $n$ unit vectors in $\mathbb{R}^r$ of their maximum pairwise inner product. In 1955, Rankin determined $\rho(r,n)$ whenever $n \leq 2r$ and in this paper we show that $\rho(r, 2r + k ) \geq \frac{\left(\frac{8}{27}k + 1\right)^{1/3} - 1}{2r + k}$, answering a question of Bukh and Cox. As a consequence, we conclude that the maximum size of a binary code with block length $r$ and minimum Hamming distance $(r-j)/2$ is at most $(2 + o(1))r$ when $j = o(r^{1/3})$, resolving a conjecture of Tiet\"av\"ainen from 1980 in a strong form. Furthermore, using a recently discovered connection to binary codes, this yields an analogous result for set-coloring Ramsey numbers of triangles.

##### 3.Improved upper bound on the Frank number of $3$-edge-connected graphs

**Authors:**János Barát, Zoltán L. Blázsik

**Abstract:** In an orientation $O$ of the graph $G$, an arc $e$ is deletable if and only if $O-e$ is strongly connected. For a $3$-edge-connected graph $G$, the Frank number is the minimum $k$ for which $G$ admits $k$ strongly connected orientations such that for every edge $e$ of $G$ the corresponding arc is deletable in at least one of the $k$ orientations. H\"orsch and Szigeti conjectured the Frank number is at most $3$ for every $3$-edge-connected graph $G$. We prove an upper bound of $5$, which improves the previous bound of $7$.

##### 4.A Schnyder-type drawing algorithm for 5-connected triangulations

**Authors:**Olivier Bernardi, Éric Fusy, Shizhe Liang

**Abstract:** We define some Schnyder-type combinatorial structures on a class of planar triangulations of the pentagon which are closely related to 5-connected triangulations. The combinatorial structures have three incarnations defined in terms of orientations, corner-labelings, and woods respectively. The wood incarnation consists in 5 spanning trees crossing each other in an orderly fashion. Similarly as for Schnyder woods on triangulations, it induces, for each vertex, a partition of the inner triangles into face-connected regions (5~regions here). We show that the induced barycentric vertex-placement, where each vertex is at the barycenter of the 5 outer vertices with weights given by the number of faces in each region, yields a planar straight-line drawing.

##### 5.Matroidal Mixed Eulerian Numbers

**Authors:**Eric Katz, Max Kutler

**Abstract:** We make a systematic study of matroidal mixed Eulerian numbers which are certain intersection numbers in the matroid Chow ring generalizing the mixed Eulerian numbers introduced by Postnikov. These numbers are shown to be valuative and obey a log-concavity relation. We establish recursion formulas and use them to relate matroidal mixed Eulerian numbers to the characteristic and Tutte polynomials, reproving results of Huh-Katz and Berget-Spink-Tseng. Generalizing Postnikov, we show that these numbers are equal to certain weighted counts of binary trees. Lastly, we study these numbers for perfect matroid designs, proving that they generalize the remixed Eulerian numbers of Nadeau-Tewari.

##### 6.A logarithmic bound for simultaneous embeddings of planar graphs

**Authors:**Raphael Steiner

**Abstract:** A set $\mathcal{G}$ of planar graphs on the same number $n$ of vertices is called simultaneously embeddable if there exists a set $P$ of $n$ points in the plane such that every graph $G \in \mathcal{G}$ admits a (crossing-free) straight-line embedding with vertices placed at points of $P$. A well-known open problem from 2007 posed by Brass, Cenek, Duncan, Efrat, Erten, Ismailescu, Kobourov, Lubiw and Mitchell, asks whether for some $n$ there exists a set $\mathcal{G}$ consisting of two planar graphs on $n$ vertices that are not simultaneously embeddable. While this remains widely open, we give a short proof that for every $\varepsilon>0$ and sufficiently large $n$ there exists a collection of at most $(4+\varepsilon)\log_2(n)$ planar graphs on $n$ vertices which cannot be simultaneously embedded. This significantly improves the previous exponential bound of $O(n\cdot 4^{n/11})$ for the same problem which was recently established by Goenka, Semnani and Yip.

##### 7.Terwilliger algebras of generalized wreath products of association schemes

**Authors:**Yuta Watanabe

**Abstract:** The generalized wreath product of association schemes was introduced by R.~A.~Bailey in European Journal of Combinatorics 27 (2006) 428--435. It is known as a generalization of both wreath and direct products of association schemes. In this paper, we discuss the Terwilliger algebra of the generalized wreath product of commutative association schemes. I will describe its structure and its central primitive idempotents in terms of the parameters of each factors and their central primitive idempotents.

##### 1.On ordered Ramsey numbers of matchings versus triangles

**Authors:**Martin Balko, Marian Poljak

**Abstract:** For graphs $G^<$ and $H^<$ with linearly ordered vertex sets, the \ordered Ramsey number $r_<(G^<,H^<)$ is the smallest positive integer $N$ such that any red-blue coloring of the edges of the complete ordered graph $K^<_N$ on $N$ vertices contains either a blue copy of $G^<$ or a red copy of $H^<$. Motivated by a problem of Conlon, Fox, Lee, and Sudakov (2017), we study the numbers $r_<(M^<,K^<_3)$ where $M^<$ is an ordered matching on $n$ vertices. We prove that almost all $n$-vertex ordered matchings $M^<$ with interval chromatic number 2 satisfy $r_<(M^<,K^<_3) \in \Omega((n/\log n)^{5/4})$ and $r_<(M^<,K^<_3) \in O(n^{7/4})$, improving a recent result by Rohatgi (2019). We also show that there are $n$-vertex ordered matchings $M^<$ with interval chromatic number at least 3 satisfying $r_<(M^<,K^<_3) \in \Omega((n/\log n)^{4/3})$, which asymptotically matches the best known lower bound on these off-diagonal ordered Ramsey numbers for general $n$-vertex ordered matchings.

##### 2.On Color Critical Graphs of Star Coloring

**Authors:**Harshit Kumar Choudhary, I. Vinod Reddy

**Abstract:** A \emph{star coloring} of a graph $G$ is a proper vertex-coloring such that no path on four vertices is $2$-colored. The minimum number of colors required to obtain a star coloring of a graph $G$ is called star chromatic number and it is denoted by $\chi_s(G)$. A graph $G$ is called $k$-critical if $\chi_s(G)=k$ and $\chi_s(G -e) < \chi_s(G)$ for every edge $e \in E(G)$. In this paper, we give a characterization of 3-critical, $(n-1)$-critical and $(n-2)$-critical graphs with respect to star coloring, where $n$ denotes the number of vertices of $G$. We also give upper and lower bounds on the minimum number of edges in $(n-1)$-critical and $(n-2)$-critical graphs.

##### 3.The characterization of infinite Eulerian graphs, a short and computable proof

**Authors:**Nicanor Carrasco-Vargas

**Abstract:** In this paper we present a short proof of a theorem by Erd\H{o}s, Gr\"unwald and Weiszfeld on the characterization of infinite graphs which admit infinite Eulerian trails. In addition, we extend this result with a characterization of which finite trails can be extended to infinite Eulerian trails. Our proof is computable and yields an effective version of this theorem. This exhibits stark contrast with other classical results in the theory of infinite graphs which are not effective.

##### 4.Branchwidth is (1,g)-self-dual

**Authors:**Georgios Kontogeorgiou, Alexandros Leivaditis, Kostas I. Psaromiligkos, Giannos Stamoulis, Dimitris Zoros

**Abstract:** A graph parameter is self-dual in some class of graphs embeddable in some surface if its value does not change in the dual graph by more than a constant factor. We prove that the branchwidth of connected hypergraphs without bridges and loops that are embeddable in some surface of Euler genus at most g is an (1,g)-self-dual parameter. This is the first proof that branchwidth is an additively self-dual width parameter.

##### 5.Non-disjoint strong external difference families can have any number of sets

**Authors:**Sophie Huczynska, Siaw-Lynn Ng

**Abstract:** Strong external difference families (SEDFs) are much-studied combinatorial objects motivated by an information security application. A well-known conjecture states that only one abelian SEDF with more than 2 sets exists. We show that if the disjointness condition is replaced by non-disjointness, then abelian SEDFs can be constructed with more than 2 sets (indeed any number of sets). We demonstrate that the non-disjoint analogue has striking differences to, and connections with, the classical SEDF and arises naturally via another coding application.

##### 6.Clones of pigmented words and realizations of special classes of monoids

**Authors:**Samuele Giraudo

**Abstract:** Clones are generalizations of operads forming powerful instruments to describe varieties of algebras wherein repeating variables are allowed in their relations. They allow us in this way to realize and study a large range of algebraic structures. A functorial construction from the category of monoids to the category of clones is introduced. The obtained clones involve words on positive integers where letters are pigmented by elements of a monoid. By considering quotients of these structures, we construct a complete hierarchy of clones involving some families of combinatorial objects. This provides clone realizations of some known and some new special classes of monoids as among others the variety of left-regular bands, bounded semilattices, and regular band monoids.

##### 7.Spectral extrema of $\{K_{k+1},\mathcal{L}_s\}$-free graphs

**Authors:**Yanni Zhai, Xiying Yuan

**Abstract:** For a set of graphs $\mathcal{F}$, a graph is said to be $\mathcal{F}$-free if it does not contain any graph in $\mathcal{F}$ as a subgraph. Let Ex$_{sp}(n,\mathcal{F})$ denote the graphs with the maximum spectral radius among all $\mathcal{F}$-free graphs of order $n$. A linear forest is a graph whose connected component is a path. Denote by $\mathcal{L}_s$ the family of all linear forests with $s$ edges. In this paper the graphs in Ex$_{sp}(n,\{K_{k+1},\mathcal{L}_s\})$ will be completely characterized when $n$ is appropriately large.

##### 8.Two sufficient conditions for graphs to admit path factors

**Authors:**Sizhong Zhou, Jiancheng Wu

**Abstract:** Let $\mathcal{A}$ be a set of connected graphs. Then a spanning subgraph $A$ of $G$ is called an $\mathcal{A}$-factor if each component of $A$ is isomorphic to some member of $\mathcal{A}$. Especially, when every graph in $\mathcal{A}$ is a path, $A$ is a path factor. For a positive integer $d\geq2$, we write $\mathcal{P}_{\geq d}=\{P_i|i\geq d\}$. Then a $\mathcal{P}_{\geq d}$-factor means a path factor in which every component admits at least $d$ vertices. A graph $G$ is called a $(\mathcal{P}_{\geq d},m)$-factor deleted graph if $G-E'$ admits a $\mathcal{P}_{\geq d}$-factor for any $E'\subseteq E(G)$ with $|E'|=m$. A graph $G$ is called a $(\mathcal{P}_{\geq d},k)$-factor critical graph if $G-Q$ has a $\mathcal{P}_{\geq d}$-factor for any $Q\subseteq V(G)$ with $|Q|=k$. In this paper, we present two degree conditions for graphs to be $(\mathcal{P}_{\geq3},m)$-factor deleted graphs and $(\mathcal{P}_{\geq3},k)$-factor critical graphs. Furthermore, we show that the two results are best possible in some sense.

##### 9.Euclidean Gallai-Ramsey for various configurations

**Authors:**Xinbu Cheng, Zixiang Xu

**Abstract:** The Euclidean Gallai-Ramsey problem, which investigates the existence of monochromatic or rainbow configurations in a colored $n$-dimensional Euclidean space $\mathbb{E}^{n}$, was introduced and studied recently. We further explore this problem for various configurations including triangles, squares, lines, and the structures with specific properties, such as rectangular and spherical configurations. Several of our new results provide refinements to the results presented in a recent work by Mao, Ozeki and Wang. One intriguing phenomenon evident on the Gallai-Ramsey results proven in this paper is that the dimensions of spaces are often independent of the number of colors. Our proofs primarily adopt a geometric perspective.

##### 10.On MaxCut and the Lovász theta function

**Authors:**Igor Balla, Oliver Janzer, Benny Sudakov

**Abstract:** In this short note we prove a lower bound for the MaxCut of a graph in terms of the Lov\'asz theta function of its complement. We combine this with known bounds on the Lov\'asz theta function of complements of $H$-free graphs to recover many known results on the MaxCut of $H$-free graphs. In particular, we give a new, very short proof of a conjecture of Alon, Krivelevich and Sudakov about the MaxCut of graphs with no cycles of length $r$.

##### 11.A new upper bound for the Heilbronn triangle problem

**Authors:**Alex Cohen, Cosmin Pohoata, Dmitrii Zakharov

**Abstract:** For sufficiently large $n$, we show that in every configuration of $n$ points chosen inside the unit square there exists a triangle of area less than $n^{-8/7-1/2000}$. This improves upon a result of Koml\'os, Pintz and Szemer\'edi from 1982. Our approach establishes new connections between the Heilbronn triangle problem and various themes in incidence geometry and projection theory which are closely related to the discretized sum-product phenomenon.

##### 12.A note on Cayley nut graphs whose degree is divisible by four

**Authors:**Ivan Damnjanović

**Abstract:** A nut graph is a non-trivial simple graph such that its adjacency matrix has a one-dimensional null space spanned by a full vector. It was recently shown by the authors that there exists a $d$-regular circulant nut graph of order $n$ if and only if $4 \mid d, \, 2 \mid n, \, d > 0$, together with $n \ge d + 4$ if $d \equiv_8 4$ and $n \ge d + 6$ if $8 \mid d$, as well as $(n, d) \neq (16, 8)$ [arXiv:2212.03026, 2022]. In this paper, we demonstrate the existence of a $d$-regular Cayley nut graph of order $n$ for each $4 \mid d, \, d > 0$ and $2 \mid n, \, n \ge d + 4$, thereby resolving the existence problem for Cayley nut graphs and vertex-transitive nut graphs whose degree is divisible by four.

##### 1.The Random Turán Problem for Theta Graphs

**Authors:**Gwen McKinley, Sam Spiro

**Abstract:** Given a graph $F$, we define $\operatorname{ex}(G_{n,p},F)$ to be the maximum number of edges in an $F$-free subgraph of the random graph $G_{n,p}$. Very little is known about $\operatorname{ex}(G_{n,p},F)$ when $F$ is bipartite, with essentially tight bounds known only when $F$ is either $C_4, C_6, C_{10}$, or $K_{s,t}$ with $t$ sufficiently large in terms of $s$, due to work of F\"uredi and of Morris and Saxton. We extend this work by establishing essentially tight bounds when $F$ is a theta graph with sufficiently many paths. Our main innovation is in proving a balanced supersaturation result for vertices, which differs from the standard approach of proving balanced supersaturation for edges.

##### 2.On the maximum of the weighted binomial sum $(1+a)^{-r}\sum_{i=0}^{r}\binom{m}{i}a^{i}$

**Authors:**Seok Hyun Byun, Svetlana Poznanović

**Abstract:** Recently, Glasby and Paseman considered the following sequence of binomial sums $\{2^{-r}\sum_{i=0}^{r}\binom{m}{i}\}_{r=0} ^{m}$ and showed that this sequence is unimodal and attains its maximum value at $r=\lfloor\frac{m}{3}\rfloor+1$ for $m\in\mathbb{Z}_{\geq0}\setminus\{0,3,6,9,12\}$. They also analyzed the asymptotic behavior of the maximum value of the sequence as $m$ approaches infinity. In the present work, we generalize their results by considering the sequence $\{(1+a)^{-r}\sum_{i=0}^{r}\binom{m}{i}a^{i}\}_{r=0} ^{m}$ for positive integers $a$. We also consider a family of discrete probability distributions that naturally arises from this sequence.

##### 1.An exponential bound for simultaneous embeddings of planar graphs

**Authors:**Ritesh Goenka, Pardis Semnani, Chi Hoi Yip

**Abstract:** We show that there are $O(n \cdot 4^{n/11})$ planar graphs on $n$ vertices which do not admit a simultaneous straight-line embedding on any $n$-point set in the plane. In particular, this improves the best known bound $O(n!)$ significantly.

##### 2.Classification des entiers monomialement irr{é}ductibles et g{é}n{é}ralisations

**Authors:**Flavien Mabilat
LMR

**Abstract:** In this article, we study the classification of some natural numbers related to the combinatorics of congruence subgroups of the modular group. More precisely, we will focus here on the notion of minimal monomial solutions. These are the solutions of a matrix equation (also appearing in the study of Coxeter friezes), modulo an integer $N$, whose components are identical and minimal for this property. Our objective here is to study the integers $N$ for which the minimal monomial solutions satisfying some fixed conditions have an irreducibility property. In particular, we will classify the monomially irreducible integers which are the integers for which all the nonzero minimal monomial solutions are irreducible.

##### 3.On the Weisfeiler-Leman dimension of permutation graphs

**Authors:**Jin Guo, Alexander L. Gavrilyuk, Ilia Ponomarenko

**Abstract:** It is proved that the Weisfeiler-Leman dimension of the class of permutation graphs is at most 18. Previously it was only known that this dimension is finite (Gru{\ss}ien, 2017).

##### 4.Proof of Nath-Sellers' Conjecture on the Largest Size of $(s,ms-1,ms+1)$-Core Partitions

**Authors:**Yetong Sha, Huan Xiong

**Abstract:** In 2016, Nath and Sellers proposed a conjecture regarding the precise largest size of ${(s,ms-1,ms+1)}$-core partitions. In this paper, we prove their conjecture. One of the key techniques in our proof is to introduce and study the properties of generalized-$\beta$-sets, which extend the concept of $\beta$-sets for core partitions. Our results can be interpreted as a generalization of the well-known result of Yang, Zhong, and Zhou concerning the largest size of $(s,s+1,s+2)$-core partitions.

##### 5.The least eigenvalue of the complements of graphs with given connectivity

**Authors:**Huan Qiu, Keng Li, Guoping Wang

**Abstract:** The least eigenvalue of a graph $G$ is the least eigenvalue of adjacency matrix of $G$. In this paper we determine the graphs which attain the minimum least eigenvalue among all complements of connected simple graphs with given connectivity.

##### 6.Linear Layouts of Bipartite Planar Graphs

**Authors:**Henry Förster, Michael Kaufmann, Laura Merker, Sergey Pupyrev, Chrysanthi Raftopoulou

**Abstract:** A linear layout of a graph $ G $ consists of a linear order $\prec$ of the vertices and a partition of the edges. A part is called a queue (stack) if no two edges nest (cross), that is, two edges $ (v,w) $ and $ (x,y) $ with $ v \prec x \prec y \prec w $ ($ v \prec x \prec w \prec y $) may not be in the same queue (stack). The best known lower and upper bounds for the number of queues needed for planar graphs are 4 [Alam et al., Algorithmica 2020] and 42 [Bekos et al., Algorithmica 2022], respectively. While queue layouts of special classes of planar graphs have received increased attention following the breakthrough result of [Dujmovi\'c et al., J. ACM 2020], the meaningful class of bipartite planar graphs has remained elusive so far, explicitly asked for by Bekos et al. In this paper we investigate bipartite planar graphs and give an improved upper bound of 28 by refining existing techniques. In contrast, we show that two queues or one queue together with one stack do not suffice; the latter answers an open question by Pupyrev [GD 2018]. We further investigate subclasses of bipartite planar graphs and give improved upper bounds; in particular we construct 5-queue layouts for 2-degenerate quadrangulations.

##### 7.A spectral radius condition for a graph to have $(a,b)$-parity factors

**Authors:**Junjie Wang, Yang Yu, Jianbiao Hu, Peng Wen

**Abstract:** Let $a,b$ be two positive integers such that $a \le b$ and $a \equiv b$ (mod $2$). We say that a graph $G$ has an $(a,b)$-parity factor if $G$ has a spanning subgraph $F$ such that $d_{F}(v) \equiv b$ (mod $2$) and $a \le d_{F}(v) \le b$ for all $v \in V (G)$. In this paper, we provide a tight spectral radius condition for a graph to have $(a,b)$-parity factors.

##### 8.A new family of $(q^4+1)$-tight sets with an automorphism group $F_4(q)$

**Authors:**Tao Feng, Weicong Li, Qing Xiang

**Abstract:** In this paper, we construct a new family of $(q^4+1)$-tight sets in $Q(24,q)$ or $Q^-(25,q)$ according as $q=3^f$ or $q\equiv 2\pmod 3$. The novelty of the construction is the use of the action of the exceptional simple group $F_4(q)$ on its minimal module over $\F_q$.

##### 9.On the $n$-attack Roman Dominating Number of a Graph

**Authors:**Garrison Koch, Nathan Shank

**Abstract:** The Roman Dominating number is a widely studied variant of the dominating number on graphs. Given a graph $G=(V,E)$, the dominating number of a graph is the minimum size of a vertex set, $V' \subseteq V$, so that every vertex in the graph is either in $V'$ or is adjacent to a vertex in $V'$. A Roman Dominating function of $G$ is defined as $f:V \rightarrow \{0,1,2\}$ such that every vertex with a label of 0 in $G$ is adjacent to a vertex with a label of 2. The Roman Dominating number of a graph is the minimum total weight over all possible Roman Dominating functions. In this paper, we analyze a new variant: $n$-attack Roman Domination, particularly focusing on 2-attack Roman Domination $(n=2)$. An $n$-attack Roman Dominating function of $G$ is defined similarly to the Roman Dominating function with the additional condition that, for any $j\leq n$, any subset $S$ of $j$ vertices all with label 0 must have at least $j$ vertices with label 2 in the open neighborhood of $S$. The $n$-attack Roman Dominating number is the minimum total weight over all possible $n$-attack Roman Dominating functions. We introduce properties as well as an algorithm to find the 2-attack Roman Domination number. We also consider how to place dominating vertices if you are only allowed a finite number of them through the help of a python program. Finally, we discuss the 2-attack Roman Dominating number of infinite regular graphs that tile the plane. We conclude with open questions and possible ways to extend these results to the general $n$-attack case.

##### 10.Tree independence number for (even hole, diamond, pyramid)-free graphs

**Authors:**Tara Abrishami, Bogdan Alecu, Maria Chudnovsky, Sepehr Hajebi, Sophie Spirkl, Kristina Vušković

**Abstract:** The tree-independence number tree-$\alpha$, first defined and studied by Dallard, Milani\v{c} and \v{S}torgel, is a variant of treewidth tailored to solving the maximum independent set problem. Over a series of papers, Abrishami et al.\ developed the so-called central bag method to study induced obstructions to bounded treewidth. Among others, they showed that, in a certain superclass $\mathcal C$ of (even hole, diamond, pyramid)-free graphs, treewidth is bounded by a function of the clique number. In this paper, we relax the bounded clique number assumption, and show that $\mathcal C$ has bounded tree-$\alpha$. Via existing results, this yields a polynomial time algorithm for the maximum independent set problem in this class. Our result also corroborates, for this class of graphs, a conjecture of Dallard, Milani\v{c} and \v{S}torgel that in a hereditary graph class, tree-$\alpha$ is bounded if and only if the treewidth is bounded by a function of the clique number.

##### 11.A study on certain bounds of the rna number and some characterizations of the parity signed graphs

**Authors:**Mohan Ramu, Joseph Varghese Kureethara

**Abstract:** For a given graph $G$, let $f:V(G)\to \{1,2,\ldots,n\}$ be a bijective mapping. For a given edge $uv \in E(G)$, $\sigma(uv)=+$, if $f(u)$ and $f(v)$ have the same parity and $\sigma(uv)=-$, if $f(u)$ and $f(v)$ have opposite parity. The resultant signed graph is called a parity signed graph. Let us denote a parity signed graph $S=(G,\sigma)$ by $G_\sigma$. Let $E^-(G_\sigma)$ be a set of negative edges in a parity signed graph and let $Si(G)$ be the set of all parity signatures for the underlying graph $G$. We define the \emph{rna} number of $G$ as $\sigma^-(G)=\min\{p(E^-(G_\sigma)):\sigma \in Si(G)\}$. In this paper, we prove a non-trivial upper bound in the case of trees: $\sigma^-(T)\leq \left \lceil \frac{n}{2}\right \rceil$, where $T$ is a tree of order $n+1$. We have found families of trees whose \emph{rna} numbers are bounded above by $\left \lceil \frac{\Delta}{2}\right \rceil$ and also we have shown that for any $i\leq \left \lceil \frac{n}{2}\right \rceil$, there exists a tree $T$ (of order $n+1$) with $\sigma^-(T)=i$. This paper gives the characterizations of graphs with \emph{rna} number 1 in terms of its spanning trees and characterizations of graphs with \emph{rna} number 2.

##### 12.Probabilistic enumeration and equivalence of nonisomorphic trees

**Authors:**Benedikt Stufler

**Abstract:** We present a new probabilistic proof of Otter's asymptotic formula for the number of unlabelled trees with a given number of vertices. We additionally prove a new approximation result, showing that the total variation distance between random P\'olya trees and random unlabelled trees tends to zero when the number of vertices tends to infinity. In order to demonstrate that our approach is not restricted to trees we extend our results to tree-like classes of graphs.

##### 13.Ordinal Sums of Numbers

**Authors:**Alexander Clow, Neil McKay

**Abstract:** In this paper we consider ordinal sums of combinatorial games where each summand is a number, not necessarily in canonical form. In doing so we give formulas for the value of an ordinal sum of numbers where the literal form of the base has certain properties. These formulas include a closed form of the value of any ordinal sum of numbers where the base is in canonical form. Our work employs a recent result of Clow which gives a criteria for an ordinal sum G : K = H : K when G and H do not have the same literal form, as well as expanding this theory with the introduction of new notation, a novel ruleset, Teetering Towers, and a novel construction of the canonical forms of numbers in Teetering Towers. In doing so, we resolve the problem of determining the value of an ordinal sum of numbers in all but a few cases appearing in Conway's On Numbers and Games; thus generalizing a number of existing results and techniques including Berlekamp' sign rule, van Roode's signed binary number method, and recent work by Carvalho, Huggan, Nowakowski, and Pereira dos Santos. We conclude with a list of open problems related to our results.

##### 14.Note on the number of antichains in generalizations of the Boolean lattice

**Authors:**Jinyoung Park, Michail Sarantis, Prasad Tetali

**Abstract:** We give a short and self-contained argument that shows that, for any positive integers $t$ and $n$ with $t =O\Bigl(\frac{n}{\log n}\Bigr)$, the number $\alpha([t]^n)$ of antichains of the poset $[t]^n$ is at most \[\exp_2\Bigl(1+O\Bigl(\Bigl(\frac{t\log^3 n}{n}\Bigr)^{1/2}\Bigr)\Bigr)N(t,n)\,,\] where $N(t,n)$ is the size of a largest level of $[t]^n$. This, in particular, says that if $t \ll n/\log^3 n$ as $n \rightarrow \infty$, then $\log\alpha([t]^n)=(1+o(1))N(t,n)$, giving a (partially) positive answer to a question of Moshkovitz and Shapira for $t, n$ in this range. Particularly for $t=3$, we prove a better upper bound: \[\log\alpha([3]^n)\le(1+4\log 3/n)N(3,n),\] which is the best known upper bound on the number of antichains of $[3]^n$.

##### 15.Sidorenko-Type Inequalities for Pairs of Trees

**Authors:**Natalie Behague, Gabriel Crudele, Jonathan A. Noel, Lina M. Simbaqueba

**Abstract:** Given two non-empty graphs $H$ and $T$, write $H\succcurlyeq T$ to mean that $t(H,G)^{|E(T)|}\geq t(T,G)^{|E(H)|}$ for every graph $G$, where $t(\cdot,\cdot)$ is the homomorphism density function. We obtain various necessary and sufficient conditions for two trees $H$ and $T$ to satisfy $H\succcurlyeq T$ and determine all such pairs on at most 8 vertices. This extends results of Leontovich and Sidorenko from the 1980s and 90s. Our approach applies an information-theoretic technique of Kopparty and Rossman to reduce the problem of showing that $H\succcurlyeq T$ for two forests $H$ and $T$ to solving a particular linear program. We also characterize trees $H$ which satisfy $H\succcurlyeq S_k$ or $H\succcurlyeq P_4$, where $S_k$ is the $k$-vertex star and $P_4$ is the $4$-vertex path.

##### 1.Rao's Theorem for forcibly planar sequences revisited

**Authors:**Riccardo W. Maffucci

**Abstract:** We consider the graph degree sequences such that every realisation is a polyhedron. It turns out that there are exactly eight of them. All of these are unigraphic, in the sense that each is realised by exactly one polyhedron. This is a revisitation of a Theorem of Rao about sequences that are realised by only planar graphs. Our proof yields additional geometrical insight on this problem. Moreover, our proof is constructive: for each graph degree sequence that is not forcibly polyhedral, we construct a non-polyhedral realisation.

##### 2.Counting oriented trees in digraphs with large minimum semidegree

**Authors:**Felix Joos, Jonathan Schrodt

**Abstract:** Let $T$ be an oriented tree on $n$ vertices with maximum degree at most $e^{o(\sqrt{\log n})}$. If $G$ is a digraph on $n$ vertices with minimum semidegree $\delta^0(G)\geq(\frac12+o(1))n$, then $G$ contains $T$ as a spanning tree, as recently shown by Kathapurkar and Montgomery (in fact, they only require maximum degree $o(n/\log n)$). This generalizes the corresponding result by Koml\'os, S\'ark\"ozy and Szemer\'edi for graphs. We investigate the natural question how many copies of $T$ the digraph $G$ contains. Our main result states that every such $G$ contains at least $|Aut(T)|^{-1}(\frac12-o(1))^nn!$ copies of $T$, which is optimal. This implies the analogous result in the undirected case.

##### 3.Shapley-Folkman-type Theorem for Integrally Convex Sets

**Authors:**Kazuo Murota, Akihisa Tamura

**Abstract:** The Shapley-Folkman theorem is a statement about the Minkowski sum of (non-convex) sets, expressing the closeness of the Minkowski sum to convexity in a quantitative manner. This paper establishes similar theorems for integrally convex sets and M-natural-convex sets, which are major classes of discrete convex sets in discrete convex analysis.

##### 4.Rooted Almost-binary Phylogenetic Networks for which the Maximum Covering Subtree Problem is Solvable in Linear Time

**Authors:**Takatora Suzuki, Han Guo, Momoko Hayamizu

**Abstract:** Phylogenetic networks are a flexible model of evolution that can represent reticulate evolution and handle complex data. Tree-based networks, which are phylogenetic networks that have a spanning tree with the same root and leaf-set as the network itself, have been well studied. However, not all networks are tree-based. Francis-Semple-Steel (2018) thus introduced several indices to measure the deviation of rooted binary phylogenetic networks $N$ from being tree-based, such as the minimum number $\delta^\ast(N)$ of additional leaves needed to make $N$ tree-based, and the minimum difference $\eta^\ast(N)$ between the number of vertices of $N$ and the number of vertices of a subtree of $N$ that shares the root and leaf set with $N$. Hayamizu (2021) has established a canonical decomposition of almost-binary phylogenetic networks of $N$, called the maximal zig-zag trail decomposition, which has many implications including a linear time algorithm for computing $\delta^\ast(N)$. The Maximum Covering Subtree Problem (MCSP) is the problem of computing $\eta^\ast(N)$, and Davidov et al. (2022) showed that this can be solved in polynomial time (in cubic time when $N$ is binary) by an algorithm for the minimum cost flow problem. In this paper, under the assumption that $N$ is almost-binary (i.e. each internal vertex has in-degree and out-degree at most two), we show that $\delta^\ast(N)\leq \eta^\ast (N)$ holds, which is tight, and give a characterisation of such phylogenetic networks $N$ that satisfy $\delta^\ast(N)=\eta^\ast(N)$. Our approach uses the canonical decomposition of $N$ and focuses on how the maximal W-fences (i.e. the forbidden subgraphs of tree-based networks) are connected to maximal M-fences in the network $N$. Our results introduce a new class of phylogenetic networks for which MCSP can be solved in linear time, which can be seen as a generalisation of tree-based networks.

##### 5.Rainbow Free Colorings and Rainbow Numbers for $x-y=z^2$

**Authors:**Katie Ansaldi, Gabriel Cowley, Eric Green, Kihyun Kim, JT Rapp

**Abstract:** An exact r-coloring of a set $S$ is a surjective function $c:S \rightarrow \{1, 2, \ldots,r\}$. A rainbow solution to an equation over $S$ is a solution such that all components are a different color. We prove that every 3-coloring of $\mathbb{N}$ with an upper density greater than $(4^s-1)/(3 \cdot 4^s)$ contains a rainbow solution to $x-y=z^k$. The rainbow number for an equation in the set $S$ is the smallest integer $r$ such that every exact $r$-coloring has a rainbow solution. We compute the rainbow numbers of $\mathbb{Z}_p$ for the equation $x-y=z^k$, where $p$ is prime and $k\geq 2$.

##### 6.Symmetry in complex unit gain graphs and their spectra

**Authors:**Pepijn Wissing, Edwin R. van Dam

**Abstract:** Complex unit gain graphs may exhibit various kinds of symmetry. In this work, we explore structural symmetry, spectral symmetry and sign-symmetry in gain graphs, and their respective relations to one-another. Our main result is a construction that transforms an arbitrary gain graph into infinitely many switching-distinct gain graphs whose spectral symmetry does not imply sign-symmetry. This provides a more general answer to the gain graph analogue of an existence question that was recently treated in the context of signed graphs.

##### 7.Strong blocking sets and minimal codes from expander graphs

**Authors:**Noga Alon, Anurag Bishnoi, Shagnik Das, Alessandro Neri

**Abstract:** A strong blocking set in a finite projective space is a set of points that intersects each hyperplane in a spanning set. We provide a new graph theoretic construction of such sets: combining constant-degree expanders with asymptotically good codes, we explicitly construct strong blocking sets in the $(k-1)$-dimensional projective space over $\mathbb{F}_q$ that have size $O( q k )$. Since strong blocking sets have recently been shown to be equivalent to minimal linear codes, our construction gives the first explicit construction of $\mathbb{F}_q$-linear minimal codes of length $n$ and dimension $k$, for every prime power $q$, for which $n = O (q k)$. This solves one of the main open problems on minimal codes.

##### 1.A matrix variant of the Erdős-Falconer distance problems over finite field

**Authors:**Hieu T. Ngo

**Abstract:** We study a matrix analog of the Erd\H{o}s-Falconer distance problems in vector spaces over finite fields. There arises an interesting analysis of certain quadratic matrix Gauss sums.

##### 2.From multivalued to Boolean functions: preservation of soft nested canalization

**Authors:**Élisabeth Remy, Paul Ruet

**Abstract:** Nested canalization (NC) is a property of Boolean functions which has been recently extended to multivalued functions. We study the effect of the Van Ham mapping (from multivalued to Boolean functions) on this property. We introduce the class of softly nested canalizing (SNC) multivalued functions, and prove that the Van Ham mapping sends SNC multivalued functions to NC Boolean functions. Since NC multivalued functions are SNC, this preservation property holds for NC multivalued functions as well. We also study the relevance of SNC functions in the context of gene regulatory network modelling.

##### 3.On the number of tangencies among 1-intersecting curves

**Authors:**Eyal Ackerman, Balázs Keszegh

**Abstract:** Let $\cal C$ be a set of curves in the plane such that no three curves in $\cal C$ intersect at a single point and every pair of curves in $\cal C$ intersect at exactly one point which is either a crossing or a touching point. According to a conjecture of J\'anos Pach the number of pairs of curves in $\cal C$ that touch each other is $O(|{\cal C}|)$. We prove this conjecture for $x$-monotone curves.

##### 4.Towards the horizons of Tits's vision -- on band schemes, crowds and F1-structures

**Authors:**Oliver Lorscheid, Koen Thas

**Abstract:** This text is dedicated to Jacques Tits's ideas on geometry over F1, the field with one element. In a first part, we explain how thin Tits geometries surface as rational point sets over the Krasner hyperfield, which links these ideas to combinatorial flag varieties in the sense of Borovik, Gelfand and White and F1-geometry in the sense of Connes and Consani. A completely novel feature is our approach to algebraic groups over F1 in terms of an alteration of the very concept of a group. In the second part, we study an incidence-geometrical counterpart of (epimorphisms to) thin Tits geometries; we introduce and classify all F1-structures on 3-dimensional projective spaces over finite fields. This extends recent work of Thas and Thas on epimorphisms of projective planes (and other rank 2 buildings) to thin planes.

##### 5.On 4-general sets in finite projective spaces

**Authors:**Francesco Pavese

**Abstract:** A $4$-general set in ${\rm PG}(n,q)$ is a set of points of ${\rm PG}(n,q)$ spanning the whole ${\rm PG}(n,q)$ and such that no four of them are on a plane. Such a pointset is said to be complete if it is not contained in a larger $4$-general set of ${\rm PG}(n, q)$. In this paper upper and lower bounds for the size of the largest and the smallest complete $4$-general set in ${\rm PG}(n,q)$, respectively, are investigated. Complete $4$-general sets in ${\rm PG}(n,q)$, $q \in \{3,4\}$, whose size is close to the theoretical upper bound are provided. Further results are also presented, including a description of the complete $4$-general sets in projective spaces of small dimension over small fields and the construction of a transitive $4$-general set of size $3(q + 1)$ in ${\rm PG}(5, q)$, $q \equiv 1 \pmod{3}$.

##### 6.The Complexity of 2-Intersection Graphs of 3-Hypergraphs Recognition for Claw-free Graphs and triangulated Claw-free Graphs

**Authors:**Niccolò Di Marco, Andrea Frosini, Christophe Picouleau

**Abstract:** Given a 3-uniform hypergraph H, its 2-intersection graph G has for vertex set the hyperedges of H and ee' is an edge of G whenever e and e' have exactly two common vertices in H. Di Marco et al. prove that deciding wether a graph G is the 2-intersection graph of a 3-uniform hypergraph is NP-complete. The main problems we study concern the class of claw-free graphs. We show that the recognition problem remains NP-complete when G is claw-free graphs but becomes polynomial if in addition G is triangulated.

##### 7.The number of topological types of trees

**Authors:**Thilo Krill, Max Pitz

**Abstract:** Two graphs are of the same topological type if they can be mutually embedded into each other topologically. We show that there are exactly $\aleph_1$ distinct topological types of countable trees. In general, for any infinite cardinal $\kappa$ there are exactly $\kappa^+$ distinct topological types of trees of size $\kappa$. This solves a problem of van der Holst from 2005.

##### 8.On generic universal rigidity on the line

**Authors:**Guilherme Zeus Dantas e Moura, Tibor Jordán, Corwin Silverman

**Abstract:** A $d$-dimensional bar-and-joint framework $(G,p)$ with underlying graph $G$ is called universally rigid if all realizations of $G$ with the same edge lengths, in all dimensions, are congruent to $(G,p)$. A graph $G$ is said to be generically universally rigid in $\mathbb{R}^d$ if every $d$-dimensional generic framework $(G,p)$ is universally rigid. In this paper we focus on the case $d=1$. We give counterexamples to a conjectured characterization of generically universally rigid graphs from R. Connelly (2011). We also introduce two new operations that preserve the universal rigidity of generic frameworks, and the property of being not universally rigid, respectively. One of these operations is used in the analysis of one of our examples, while the other operation is applied to obtain a lower bound on the size of generically universally rigid graphs. This bound gives a partial answer to a question from T. Jord\'an and V-H. Nguyen (2015).

##### 9.Settling the nonorientable genus of the nearly complete bipartite graphs

**Authors:**Warren Singh, Timothy Sun

**Abstract:** A graph is said to be nearly complete bipartite if it can be obtained by deleting a set of independent edges from a complete bipartite graph. The nonorientable genus of such graphs is known except in a few cases where the sizes of the partite classes differ by at most one, and a maximum matching is deleted. We resolve these missing cases using three classic tools for constructing genus embeddings of the complete bipartite graphs: current graphs, diamond sums, and the direct rotation systems of Ringel.

##### 10.Equitable Choosability of Prism Graphs

**Authors:**Kirsten Hogenson, Dan Johnston, Suzanne O'Hara

**Abstract:** A graph $G$ is equitably $k$-choosable if, for every $k$-uniform list assignment $L$, $G$ is $L$-colorable and each color appears on at most $\left\lceil |V(G)|/k\right\rceil$ vertices. Equitable list-coloring was introduced by Kostochka, Pelsmajer, and West in 2003. They conjectured that a connected graph $G$ with $\Delta(G)\geq 3$ is equitably $\Delta(G)$-choosable, as long as $G$ is not complete or $K_{d,d}$ for odd $d$. In this paper, we use a discharging argument to prove their conjecture for the infinite family of prism graphs.

##### 11.Network Routing on Regular Digraphs and Their Line Graphs

**Authors:**Vance Faber, Noah Streib

**Abstract:** This paper concerns all-to-all network routing on regular digraphs. In previous work we focused on efficient routing in highly symmetric digraphs with low diameter for fixed degree. Here, we show that every connected regular digraph has an all-to-all routing scheme and associated schedule with no waiting. In fact, this routing scheme becomes more efficient as the diameter goes down with respect to the degree and number of vertices. Lastly, we examine the simple scheduling algorithm called ``farthest-distance-first'' and prove that it yields optimal schedules for all-to-all communication in networks of interest, including Kautz graphs.

##### 12.Set-coloring Ramsey numbers and error-correcting codes near the zero-rate threshold

**Authors:**David Conlon, Jacob Fox, Huy Tuan Pham, Yufei Zhao

**Abstract:** For positive integers $n,r,s$ with $r > s$, the set-coloring Ramsey number $R(n;r,s)$ is the minimum $N$ such that if every edge of the complete graph $K_N$ receives a set of $s$ colors from a palette of $r$ colors, then there is a subset of $n$ vertices where all of the edges between them receive a common color. If $n$ is fixed and $\frac{s}{r}$ is less than and bounded away from $1-\frac{1}{n-1}$, then $R(n;r,s)$ is known to grow exponentially in $r$, while if $\frac{s}{r}$ is greater than and bounded away from $1-\frac{1}{n-1}$, then $R(n;r,s)$ is bounded. Here we prove bounds for $R(n;r,s)$ in the intermediate range where $\frac{s}{r}$ is close to $1 - \frac{1}{n-1}$ by establishing a connection to the maximum size of error-correcting codes near the zero-rate threshold.

##### 13.Bijective enumeration of general stacks

**Authors:**Qianghui Guo, Yinglie Jin, Lisa H. Sun, Shina Xu

**Abstract:** Combinatorial enumeration of various RNA secondary structures and protein contact maps, is of great interest for both combinatorists and computational biologists. Enumeration of protein contact maps has considerable difficulties due to the significant higher vertex degree than that of RNA secondary structures. The state of art maximum vertex degree in previous works is two. This paper proposes a solution for counting stacks in protein contact maps with arbitrary vertex degree upper bound. By establishing bijection between such general stacks and $m$-regular $\Lambda$-avoiding $DLU$ paths, and counting the paths using theories of pattern avoiding lattice paths, we obtain a unified system of equations for generating functions of general stacks. We also show that previous enumeration results for RNA secondary structures and protein contact maps can be derived from the unified equation system as special cases.

##### 14.Non-uniform skew versions of Bollobás' Theorem

**Authors:**Gábor Hegedüs

**Abstract:** Let $A_1, \ldots ,A_m$ and $B_1, \ldots ,B_m$ be subsets of $[n]$ and let $t$ be a non-negative integer with the following property: $|A_i \cap B_i|\leq t$ for each $i$ and $|A_i\cap B_j|>t$ whenever $i< j$. Then $m\leq 2^{n-t}$. Our proof uses Lov\'asz' tensor product method. We prove the following skew version of Bollob\'as' Theorem. Let $A_1, \ldots ,A_m$ and $B_1, \ldots ,B_m$ be finite sets of $[n]$ satisfying the conditions $A_i \cap B_i =\emptyset$ for each $i$ and $A_i\cap B_j\ne \emptyset$ for each $i< j$. Then $$ \sum_{i=1}^m \frac{1}{{|A_i|+|B_i| \choose |A_i|}}\leq n+1. $$ Both upper bounds are sharp.

##### 15.Pretty good state transfer among large sets of vertices

**Authors:**Ada Chan, Peter Sin

**Abstract:** In a continuous-time quantum walk on a network of qubits, pretty good state transfer is the phenomenon of state transfer between two vertices with fidelity arbitrarily close to 1. We construct families of graphs to demonstrate that there is no bound on the size of a set of vertices that admit pretty good state transfer between any two vertices of the set.

##### 1.Minimum degree conditions for rainbow triangles

**Authors:**Victor Falgas-Ravry, Klas Markström, Eero Räty

**Abstract:** Let $\mathbf{G}:=(G_1, G_2, G_3)$ be a triple of graphs on a common vertex set $V$ of size $n$. A rainbow triangle in $\mathbf{G}$ is a triple of edges $(e_1, e_2, e_3)$ with $e_i\in G_i$ for each $i$ and $\{e_1, e_2, e_3\}$ forming a triangle in $V$. In this paper we consider the following question: what triples of minimum degree conditions $(\delta(G_1), \delta(G_2), \delta(G_3))$ guarantee the existence of a rainbow triangle? This may be seen as a minimum degree version of a problem of Aharoni, DeVos, de la Maza, Montejanos and \v{S}\'amal on density conditions for rainbow triangles, which was recently resolved by the authors. We establish that the extremal behaviour in the minimum degree setting differs strikingly from that seen in the density setting, with discrete jumps as opposed to continuous transitions. Our work leaves a number of natural questions open, which we discuss.

##### 2.Finite matchability under the matroidal Hall's condition

**Authors:**Attila Joó

**Abstract:** Aharoni and Ziv conjectured that if $ M $ and $ N $ are finitary matroids on $ E $, then a certain ``Hall-like'' condition is sufficient to guarantee the existence of an $ M $-independent spanning set of $ N $. We show that their condition ensures that every finite subset of $ E $ is $ N $-spanned by an $ M $-independent set.

##### 3.On reduction for eigenfunctions of graphs

**Authors:**Alexandr Valyuzhenich

**Abstract:** In this work, we prove a general version of the reduction lemmas for eigenfunctions of graphs admitting involutive automorphisms of a special type.

##### 4.On the edge-Erdős-Pósa property of walls

**Authors:**Raphael Steck

**Abstract:** I show that walls of size at least $6 \times 4$ do not have the edge-Erd\H{o}s-P\'{o}sa property.

##### 5.Moore-Penrose inverse of incidence matrices

**Authors:**Ali Azimi, R. B. Bapat, Mohammad Farrokhi Derakhshandeh Ghouchan

**Abstract:** We present explicit formulas for Moore-Penrose inverses of some families of set inclusion matrices arising from sets, vector spaces, and designs.

##### 6.On bridge graphs with local antimagic chromatic number 3

**Authors:**W. C. Shiu, G. C. Lau, R. X. Zhang

**Abstract:** Let $G=(V, E)$ be a connected graph. A bijection $f: E\to \{1, \ldots, |E|\}$ is called a local antimagic labeling if for any two adjacent vertices $x$ and $y$, $f^+(x)\neq f^+(y)$, where $f^+(x)=\sum_{e\in E(x)}f(e)$ and $E(x)$ is the set of edges incident to $x$. Thus a local antimagic labeling induces a proper vertex coloring of $G$, where the vertex $x$ is assigned the color $f^+(x)$. The local antimagic chromatic number $\chi_{la}(G)$ is the minimum number of colors taken over all colorings induced by local antimagic labelings of $G$. In this paper, we present some families of bridge graphs with $\chi_{la}(G)=3$ and give several ways to construct bridge graphs with $\chi_{la}(G)=3$.

##### 7.Uniqueness of an association scheme related to the Witt design on 11 points

**Authors:**Alexander L. Gavrilyuk, Sho Suda

**Abstract:** It follows from Delsarte theory that the Witt $4$-$(11,5,1)$ design gives rise to a $Q$-polynomial association scheme $\mathcal{W}$ defined on the set of its blocks. In this note we show that $\mathcal{W}$ is unique, i.e., defined up to isomorphism by its parameters.

##### 8.Some Separable integer partition classes

**Authors:**Y. H. Chen, Thomas Y. He, F. Tang, J. J. Wei

**Abstract:** In this paper, we investigate partitions with parts separated by parity introduced by Andrews with the aid of separation integer partition classes. we also extend separable integer partition classes with modulus $1$ introduced by Andrews to overpartitions, called separable overpartition classes. We investigate overpartition and the overpartition analogue of Rogers-Ramanujan indentities, which are separable overpartition classes.

##### 1.Complexity of Near-3-Choosability Problem

**Authors:**Sounaka Mishra, Rohini S, Sagar S. Sawant

**Abstract:** It is currently an unsolved problem to determine whether a $\triangle$-free planar graph $G$ contains an independent set $A$ such that $G[V_G\setminus A]$ is $2$-choosable. However, in this paper, we take a slightly different approach by relaxing the planarity condition. We prove the $\mathbb{NP}$-completeness of the above decision problem when the graph is $\triangle$-free, $4$-colorable, and of diameter $3$. Building upon this notion, we examine the computational complexity of two optimization problems: minimum near $3$-choosability and minimum $2$-choosable deletion. In the former problem, the goal is to find an independent set $A$ of minimum size in a given graph $G$, such that the induced subgraph $G[V_G \setminus A]$ is $2$-choosable. We establish that this problem is $\mathbb{NP}$-hard to approximate within a factor of $|V_G|^{1-\epsilon}$ for any $\epsilon > 0$, even for planar bipartite graphs. On the other hand, the problem of minimum $2$-choosable deletion involves determining a vertex set $A \subseteq V_G$ of minimum cardinality such that the induced subgraph $G[V_G \setminus A]$ is $2$-choosable. We prove that this problem is $\mathbb{NP}$-complete, but can be approximated within a factor of $O(\log |V_G|)$.

##### 2.Colorings of some Cayley graphs

**Authors:**Prajnanaswaroop S

**Abstract:** Cayley graphs are graphs on algebraic structures, typically groups or group-like structures. In this paper, we have obtained a few results on Cayley graphs on Cyclic groups, typically powers of cycles, some colorings of powers of cycles, Cayley graphs on some non-abelian groups, and Cayley graphs on gyrogroups.

##### 3.The Turán number of special linear forest and star-path forest

**Authors:**Xiaona Fang, Lihua You

**Abstract:** The Tur\'an number of a graph $F$, denoted $ex(n, F)$, is the maximum number of edges in an $F$-free graph on $n$ vertices. Let $P_{\ell}$, $S_{\ell}$ denote the path and star on $\ell$ vertices, respectively. A linear forest is a forest whose connected components are paths. In 2013, Lidick\'y et al. considered the Tur\'an number of linear forest and $k_1P_4 \bigcup k_2 S_3$ for sufficiently large $n$. Recently, Fang and Yuan determine the Tur\'an numbers of $P_{\ell} \bigcup kS_{\ell-1}$, $k_1P_{2\ell} \bigcup k_2 S_{2\ell-1}$, $2P_5 \bigcup kS_4$ for $n$ appropriately large and characterized the corresponding extremal graphs. In this paper, We determine $ex(n, P_9 \bigcup P_7)$ for all $n\geq 16$ and characterize all extremal graphs, which partially confirms a conjecture proposed by Yuan and Zhang [L.T. Yuan, X.D. Zhang, J. Graph. Theory 98(3) (2021) 499--524]. And we determine the Tur\'an numbers of $\bigcup\limits _{i=1}^{k_1} P_{\ell _i} \bigcup\limits _{j=1}^{k_2} S_{a _j}$ for $n$ appropriately large, where $a_j\leq 2k_2+2 \sum\limits_{i=1}^{k_1} \lfloor\frac{\ell_i}{2}\rfloor-2j$ for any $j\in [k_2]$, which generalizes the results of Fang and Yuan. The corresponding extremal graphs are also completely characterized

##### 4.Towards inductive proofs in algebraic combinatorics

**Authors:**Ted Dobson

**Abstract:** We introduce a new class of transitive permutation groups which properly contains the automorphism groups of vertex-transitive graphs and digraphs. We then give a sufficient condition for a quotient of this family to remain in the family, showing that relatively straightforward induction arguments may possibly be used to solve problems in this family, and consequently for symmetry questions about vertex-transitive digraphs. As an example of this, for $p$ an odd prime, we use induction to determine the Sylow $p$-subgroups of transitive groups of degree $p^n$ that contain a regular cyclic subgroup in this family. This is enough information to determine the automorphism groups of circulant digraphs of order $p^n$.

##### 5.Skew symplectic and orthogonal characters through lattice paths

**Authors:**Seamus P. Albion, Ilse Fischer, Hans Höngesberg, Florian Schreier-Aigner

**Abstract:** The skew Schur functions admit many determinantal expressions. Chief among them are the (dual) Jacobi-Trudi formula and Lascoux-Pragacz formula, which is a skew analogue of the Giambelli identity. Comparatively, the skew characters of the symplectic and orthogonal groups, also known as the skew symplectic and orthogonal Schur functions, have received very little attention in this direction. We establish analogues of the dual Jacobi-Trudi and Lascoux-Pragacz formulae for these characters. Our approach is entirely combinatorial, being based on lattice path descriptions of the tableaux models of Koike and Terada.

##### 6.The $Z_q$-forcing number for some graph families

**Authors:**Jorge Blanco, Stephanie Einstein, Caleb Hostetler, Jurgen Kritschgau, Daniel Ogbe

**Abstract:** The zero forcing number was introduced as a combinatorial bound on the maximum nullity taken over the set of real symmetric matrices that respect the pattern of an underlying graph. The $Z_q$-forcing game is an analog to the standard zero forcing game which incorporates inertia restrictions on the set of matrices associated with a graph. This work proves an upper bound on the $Z_q$-forcing number for trees. Furthermore, we consider the $Z_q$-forcing number for caterpillar cycles on $n$ vertices. We focus on developing game theoretic proofs of upper and lower bounds.

##### 7.Hypergraphs with a quarter uniform Turán density

**Authors:**Hao Li, Hao Lin, Guanghui Wang, Wenling Zhou

**Abstract:** The uniform Tur\'an density $\pi_{1}(F)$ of a $3$-uniform hypergraph $F$ is the supremum over all $d$ for which there is an $F$-free hypergraph with the property that every linearly sized subhypergraph with density at least $d$. Determining $\pi_{1}(F)$ for given hypergraphs $F$ was suggested by Erd\H{o}s and S\'os in 1980s. In particular, they raised the questions of determining $\pi_{1}(K_4^{(3)-})$ and $\pi_{1}(K_4^{(3)})$. The former question was solved recently in [Israel J. Math. 211 (2016), 349-366] and [J. Eur. Math. Soc. 20 (2018), 1139-1159], while the latter is still a major open problem. In addition to $K_4^{(3)-}$, there are very few hypergraphs whose uniform Tur\'an density has been determined. In this paper, we give a sufficient condition for $3$-uniform hypergraphs $F$ satisfying $\pi_{1}(F)=1/4$. In particular, currently all known $3$-uniform hypergraphs whose uniform Tur\'an density is $1/4$, such as $K_4^{(3)-}$ and the $3$-uniform hypergraphs $F^{\star}_5$ studied in [arXiv:2211.12747], satisfy this condition. Moreover, we find some intriguing $3$-uniform hypergraphs whose uniform Tur\'an density is also $1/4$.

##### 8.Linear $χ$-binding functions for $\{P_3\cup P_2, gem\}$-free graphs

**Authors:**Athmakoori Prashant, S. Francis Raj, M. Gokulnath

**Abstract:** Finding families that admit a linear $\chi$-binding function is a problem that has interested researchers for a long time. Recently, the question of finding linear subfamilies of $2K_2$-free graphs has garnered much attention. In this paper, we are interested in finding a linear subfamily of a specific superclass of $2K_2$-free graphs, namely $(P_3\cup P_2)$-free graphs. We show that the class of $\{P_3\cup P_2,gem\}$-free graphs admits $f=2\omega$ as a linear $\chi$-binding function. Furthermore, we give examples to show that the optimal $\chi$-binding function $f^*\geq \left\lceil\frac{5\omega(G)}{4}\right\rceil$ for the class of $\{P_3\cup P_2, gem\}$-free graphs and that the $\chi$-binding function $f=2\omega$ is tight when $\omega=2$ and $3$.

##### 9.Bisector fields of quadrilaterals

**Authors:**Bruce Olberding, Elaine A. Walker

**Abstract:** Working over a field of characteristic other than $2$, we examine a relationship between quadrilaterals and the pencil of conics passing through their vertices. Asymptotically, such a pencil of conics is what we call a bisector field, a set ${\mathbb{B}}$ of paired lines such that each line $\ell$ in ${\mathbb{B}}$ simultaneously bisects each pair in ${\mathbb{B}}$ in the sense that $\ell$ crosses the pairs of lines in ${\mathbb{B}}$ in pairs of points that all share the same midpoint. We show that a quadrilateral induces a geometry on the affine plane via an inner product, under which we examine pencils of conics and pairs of bisectors of a quadrilateral. We show also how bisectors give a new interpretation of some classically studied features of quadrangles, such as the nine-point conic.

##### 10.Cliques in Squares of Graphs with Maximum Average Degree less than 4

**Authors:**Daniel W. Cranston, Gexin Yu

**Abstract:** Hocquard, Kim, and Pierron constructed, for every even integer $D\ge 2$, a 2-degenerate graph $G_D$ with maximum degree $D$ such that $\omega(G_D^2)=\frac52D$. They asked whether (a) there exists $D_0$ such that every 2-degenerate graph $G$ with maximum degree $D\ge D_0$ satisfies $\chi(G^2)\le \frac52D$ and (b) whether this result holds more generally for every graph $G$ with mad(G)<4. In this direction, we prove upper bounds on the clique number $\omega(G^2)$ of $G^2$ that match the lower bound given by this construction, up to small additive constants. We show that if $G$ is 2-degenerate with maximum degree $D$, then $\omega(G^2)\le \frac52D+72$ (with $\omega(G^2)\le \frac52D+60$ when $D$ is sufficiently large). And if $G$ has mad(G)<4 and maximum degree $D$, then $\omega(G^2)\le \frac52D+532$. Thus, the construction of Hocquard et al. is essentially best possible.

##### 11.Colored Permutation Statistics by Conjugacy Class

**Authors:**Jesse Campion Loth, Michael Levet, Kevin Liu, Sheila Sundaram, Mei Yin

**Abstract:** In this paper, we consider the moments of permutation statistics on conjugacy classes of colored permutation groups. We first show that when the cycle lengths are sufficiently large, the moments of arbitrary permutation statistics are independent of the conjugacy class. For permutation statistics that can be realized via $\textit{symmetric}$ constraints, we show that for a fixed number of colors, each moment is a polynomial in the degree $n$ of the $r$-colored permutation group $\mathfrak{S}_{n,r}$. Hamaker & Rhoades (arXiv 2022) established analogous results for the symmetric group as part of their far-reaching representation-theoretic framework. Independently, Campion Loth, Levet, Liu, Stucky, Sundaram, & Yin (arXiv, 2023) arrived at independence and polynomiality results for the symmetric group using instead an elementary combinatorial framework. Our techniques in this paper build on this latter elementary approach.

##### 12.Cayley extensions of maniplexes and polytopes

**Authors:**Gabe Cunningham
Wentworth Institute of Technology, Elías Mochán
Northeastern University, Antonio Montero
University of Ljubljana

**Abstract:** A map on a surface whose automorphism group has a subgroup acting regularly on its vertices is called a Cayley map. Here we generalize that notion to maniplexes and polytopes. We define $\mathcal{M}$ to be a \emph{Cayley extension} of $\mathcal{K}$ if the facets of $\mathcal{M}$ are isomorphic to $\mathcal{K}$ and if some subgroup of the automorphism group of $\mathcal{M}$ acts regularly on the facets of $\mathcal{M}$. We show that many natural extensions in the literature on maniplexes and polytopes are in fact Cayley extensions. We also describe several universal Cayley extensions. Finally, we examine the automorphism group and symmetry type graph of Cayley extensions.

##### 13.New classes of groups related to algebraic combinatorics with applications to isomorphism problems

**Authors:**Ted Dobson

**Abstract:** We introduce two refinements of the class of $5/2$-groups, inspired by the classes of automorphism groups of configurations and automorphism groups of unit circulant digraphs. We show that both of these classes have the property that any two regular cyclic subgroups of a group $G$ in either of these classes are conjugate in $G$. This generalizes two results in the literature (and simplifies their proofs) that show that symmetric configurations and unit circulant digraphs are isomorphic if and only if they are isomorphic by a group automorphism of ${\mathbb Z}_n$.

##### 1.Modified ascent sequences and Bell numbers

**Authors:**Giulio Cerbai

**Abstract:** In 2011, Duncan and Steingr\'imsson conjectured that modified ascent sequences avoiding any of the patterns 212, 1212, 2132, 2213, 2231 and 2321 are counted by the Bell numbers. Furthermore, the distribution of the number of ascents is the reverse of the distribution of blocks on set partitions. We solve the conjecture for all the patterns except 2321. We describe the corresponding sets of Fishburn permutations by pattern avoidance, and leave some open questions for future work.

##### 2.The Generalized Combinatorial Lason-Alon-Zippel-Schwartz Nullstellensatz Lemma

**Authors:**Günter Rote

**Abstract:** We survey a few strengthenings and generalizations of the Combinatorial Nullstellensatz of Alon and the Schwartz-Zippel Lemma. These lemmas guarantee the existence of (a certain number of) nonzeros of a multivariate polynomial when the variables run independently through sufficiently large ranges.

##### 3.Strongly common graphs with odd girth are cycles

**Authors:**Leo Versteegen

**Abstract:** A graph $H$ is called strongly common if for every coloring $\phi$ of $K_n$ with two colors, the number of monochromatic copies of $H$ is at least the number of monochromatic copies of $H$ in a random coloring of $K_n$ with the same density of color classes as $\phi$. In this note we prove that if a graph has odd girth but is not a cycle, then it is not strongly common. This answers a question of Chen and Ma.

##### 4.Weak saturation in graphs: a combinatorial approach

**Authors:**Nikolay Terekhov, Maksim Zhukovskii

**Abstract:** The weak saturation number $\mathrm{wsat}(n,F)$ is the minimum number of edges in a graph on $n$ vertices such that all the missing edges can be activated sequentially so that each new edge creates a copy of $F$. A usual approach to prove a lower bound for the weak saturation number is algebraic: if it is possible to embed edges of $K_n$ in a vector space in a certain way (depending on $F$), then the dimension of the subspace spanned by the images of the edges of $K_n$ is a lower bound for the weak saturation number. In this paper, we present a new combinatorial approach to prove lower bounds for weak saturation numbers that allows to establish worst-case tight (up to constant additive terms) general lower bounds as well as to get exact values of the weak saturation numbers for certain graph families. It is known (Alon, 1985) that, for every $F$, there exists $c_F$ such that $\mathrm{wsat}(n,F)=c_Fn(1+o(1))$. Our lower bounds imply that all values in the interval $\left[\frac{\delta}{2}-\frac{1}{\delta+1},\delta-1\right]$ with step size $\frac{1}{\delta+1}$ are achievable by $c_F$ (while any value outside this interval is not achievable).

##### 5.Reconstructing a set from its subset sums: $2$-torsion-free groups

**Authors:**Federico Glaudo, Noah Kravitz

**Abstract:** For a finite multiset $A$ of an abelian group $G$, let $\text{FS}(A)$ denote the multiset of the $2^{|A|}$ subset sums of $A$. It is natural to ask to what extent $A$ can be reconstructed from $\text{FS}(A)$. We fully solve this problem for $2$-torsion-free groups $G$ by giving characterizations, both algebraic and combinatorial, of the fibers of $\text{FS}$. Equivalently, we characterize all pairs of multisets $A,B$ with $\text{FS}(A)=\text{FS}(B)$. Our results build on recent work of Ciprietti and the first author.

##### 6.Ranges of polynomials control degree ranks of Green and Tao over finite prime fields

**Authors:**Thomas Karam

**Abstract:** Let $p$ be a prime, let $1 \le t < d < p$ be integers, and let $S$ be a non-empty subset of $\mathbb{F}_p$. We establish that if a polynomial $P:\mathbb{F}_p^n \to \mathbb{F}_p$ with degree $d$ is such that the image $P(S^n)$ does not contain the full image $A(\mathbb{F}_p)$ of any non-constant polynomial $A: \mathbb{F}_p \to \mathbb{F}_p$ with degree at most $t$, then $P$ coincides on $S^n$ with a polynomial that in particular has bounded degree-$\lfloor d/(t+1) \rfloor$-rank in the sense of Green and Tao. Similarly, we prove that if the assumption holds even for $t=d$, then $P$ coincides on $S^n$ with a polynomial determined by a bounded number of coordinates.

##### 1.Further Results on Random Walk Labelings

**Authors:**Sela Fried, Toufik Mansour

**Abstract:** Recently, we initiated the study of random walk labelings of graphs. These are graph labelings that are obtainable by performing a random walk on the graph, such that each vertex is labeled upon its first visit. In this work, we calculate the number of random walk labelings of several natural graph families: The wheel, fan, barbell, lollipop, tadpole, friendship, and snake graphs. Additionally, we prove several combinatorial identities that emerged during the calculations.

##### 2.Constructing nonorientable genus embedding of complete bipartite graph minus a matching

**Authors:**Shengxiang Lv

**Abstract:** $G_{m,n,k}$ is a subgraph of the complete bipartite graph $K_{m,n}$ with a $k$-matching removed. By a new method based on the embeddings of some $G_{m,n,k}$ with small $m,n,k$ and bipartite joins with small bipartite graphs, we construct the nonorientable genus embedding of $G_{m,n,k}$ for all $m,n\geq 3$ with $(m,n,k)\neq (5,4,4), (4,5,4),(5,5,5)$. Hence, we solve the cases $G_{n+1,n,n}$($n$ is even) and $G_{n,n,n}$, with the values of $n$ that have not been previously solved, i.e., $n\geq 6$. This completes previous work on the nonorientable genus of $G_{m,n,k}$.

##### 3.Generalized Polymorphisms

**Authors:**Gilad Chase, Yuval Filmus

**Abstract:** We determine all $m$-ary Boolean functions $f_0,\ldots,f_m$ and $n$-ary Boolean functions $g_0,\ldots,g_n$ satisfying the equation \[ f_0(g_1(z_{11},\ldots,z_{1m}),\ldots,g_n(z_{n1},\ldots,z_{nm})) = g_0(f_1(z_{11},\ldots,z_{n1}),\ldots,f_m(z_{1m},\ldots,z_{nm})), \] for all Boolean inputs $\{ z_{ij} : i \in [n], j \in [m] \}$. This extends characterizations by Dokow and Holzman (who considered the case $g_0 = \cdots = g_n$) and by Chase, Filmus, Minzer, Mossel and Saurabh (who considered the case $g_1 = \cdots = g_n$).

##### 4.Inverse problem for electrical networks via twist

**Authors:**Terrence George

**Abstract:** We construct an electrical-network version of the twist map for the positive Grassmannian, and use it to solve the inverse problem of recovering conductances from the response matrix. Each conductance is expressed as a biratio of Pfaffians as in the inverse map of Kenyon and Wilson; however, our Pfaffians are the more canonical $B$ variables instead of their tripod variables, and are coordinates on the positive orthogonal Grassmannian studied by Henriques and Speyer.

##### 5.On the extremal families for the Kruskal--Katona theorem

**Authors:**Oriol Serra, Lluís Vena

**Abstract:** In \cite[Serra, Vena, Extremal families for the Kruskal-Katona theorem]{sv21}, the authors have shown a characterization of the extremal families for the Kruskal-Katona Theorem. We further develop some of the arguments given in \cite{sv21} and give additional properties of these extremal families. F\"uredi-Griggs/M\"ors theorem from 1986/85 \cite{furgri86,mors85} claims that, for some cardinalities, the initial segment of the colexicographical is the unique extremal family; we extend their result as follows: the number of (non-isomorphic) extremal families strictly grows with the gap between the last two coefficients of the $k$-binomial decomposition. We also show that every family is an induced subfamily of an extremal family, and that, somewhat going in the opposite direction, every extremal family is close to being the inital segment of the colex order; namely, if the family is extremal, then after performing $t$ lower shadows, with $t=O(\log(\log n))$, we obtain the initial segment of the colexicographical order. We also give a ``fast'' algorithm to determine whether, for a given $t$ and $m$, there exists an extremal family of size $m$ for which its $t$-th lower shadow is not yet the initial segment in the colexicographical order. As a byproduct of these arguments, we give yet another characterization of the families of $k$-sets satisfying equality in the Kruskal--Katona theorem. Such characterization is, at first glance, less appealing than the one in \cite{sv21}, since the additional information that it provides is indirect. However, the arguments used to prove such characterization provide additional insight on the structure of the extremal families themselves.

##### 6.Orienting undirected phylogenetic networks to tree-child network

**Authors:**Shunsuke Maeda, Yusuke Kaneko, Hideaki Muramatsu, Yukihiro Murakami, Momoko Hayamizu

**Abstract:** Phylogenetic networks are used to represent the evolutionary history of species. They are versatile when compared to traditional phylogenetic trees, as they capture more complex evolutionary events such as hybridization and horizontal gene transfer. Distance-based methods such as the Neighbor-Net algorithm are widely used to compute phylogenetic networks from data. However, the output is necessarily an undirected graph, posing a great challenge to deduce the direction of genetic flow in order to infer the true evolutionary history. Recently, Huber et al. investigated two different computational problems relevant to orienting undirected phylogenetic networks into directed ones. In this paper, we consider the problem of orienting an undirected binary network into a tree-child network. We give some necessary conditions for determining the tree-child orientability, such as a tight upper bound on the size of tree-child orientable graphs, as well as many interesting examples. In addition, we introduce new families of undirected phylogenetic networks, the jellyfish graphs and ladder graphs, that are orientable but not tree-child orientable. We also prove that any ladder graph can be made tree-child orientable by adding extra leaves, and describe a simple algorithm for orienting a ladder graph to a tree-child network with the minimum number of extra leaves. We pose many open problems as well.

##### 7.Epimorphisms of generalized polygons B: The octagons

**Authors:**Joseph A. Thas, Koen Thas

**Abstract:** This is the second part of our study of epimorphisms with source a thick generalized $m$-gon and target a thin generalized $m$-gon. We classify the case $m = 8$ when the polygons are finite (in the first part [15] we handled the cases $m = 3, 4$ and $6$). Then we show that the infinite case is very different, and construct examples which strongly differ from the finite case. A number of general structure theorems are also obtained, and we also take a look at the infinite case for general gonality.

##### 8.No perfect state transfer in trees with more than 3 vertices

**Authors:**Gabriel Coutinho, Emanuel Juliano, Thomás Jung Spier

**Abstract:** We prove that the only trees that admit perfect state transfer according to the adjacency matrix model are $P_2$ and $P_3$. This answers a question first asked by Godsil in 2012 and proves a conjecture by Coutinho and Liu from 2015.

##### 9.Addition-deletion theorems for the Solomon-Terao polynomials and $B$-sequences of hyperplane arrangements

**Authors:**Takuro Abe

**Abstract:** We prove the addition-deletion theorems for the Solomon-Terao polynomials, which have two important specializations. Namely, one is to the characteristic polynomials of hyperplane arangements, and the other to the Poincar\`{e} polynomials of the regular nilpotent Hessenberg varieties. One of the main tools to show them is the free surjection theorem which confirms the right exactness of several important exact sequences among logarithmic modules. Moreover, we introduce a generalized polynomial $B$-theory to the higher order logarithmic modules, whose origin was due to Terao.

##### 10.Unsolved Problems in Spectral Graph Theory

**Authors:**Lele Liu, Bo Ning

**Abstract:** Spectral graph theory is a captivating area of graph theory that employs the eigenvalues and eigenvectors of matrices associated with graphs to study them. In this paper, we present a collection of $20$ topics in spectral graph theory, covering a range of open problems and conjectures. Our focus is primarily on the adjacency matrix of graphs, and for each topic, we provide a brief historical overview.

##### 1.Decomposition of (infinite) digraphs along directed 1-separations

**Authors:**Nathan Bowler, Florian Gut, Meike Hatzel, Ken-ichi Kawarabayashi, Irene Muzi, Florian Reich

**Abstract:** We introduce torsoids, a canonical structure in matching covered graphs, corresponding to the bricks and braces of the graph. This allows a more fine-grained understanding of the structure of finite and infinite directed graphs with respect to their 1-separations.

##### 2.Major index on involutions

**Authors:**Eli Bagno, Yisca Kares

**Abstract:** We find the range of the major index on the various conjugacy classes of involutions in the symmetric group $S_n$. In addition to indicating the minimum and the maximum values, we show that except for the case of involutions without fixed points, all the values in the range are attained. For the conjugacy classes of involutions without fixed points, we show that the only missing values are one more than the minimum and one less than the maximum.

##### 3.The spectrum of symmetric decorated paths

**Authors:**Gabriel Coutinho, Emanuel Juliano, Thomás Jung Spier

**Abstract:** The main result of this paper states that in a rooted product of a path with rooted graphs which are disposed in a somewhat mirror-symmetric fashion, there are distinct eigenvalues supported in the end vertices of the path which are too close to each other: their difference is smaller than the square root of two in the even distance case, and smaller than one in the odd distance case. As a first application, we show that these end vertices cannot be involved in a quantum walk phenomenon known as perfect state transfer, significantly strengthening a recent result by two of the authors along with Godsil and van Bommel. For a second application, we show that there is no balanced integral tree of odd diameter bigger than three, answering a question raised by H\'{i}c and Nedela in 1998. Our main technique involves manipulating ratios of characteristic polynomials of graphs and subgraphs into continued fractions, and exploring in detail their analytic properties. We will also make use of a result due to P\'{o}lya and Szeg\"{o} about functions that preserve the Lebesgue measure, which as far as we know is a novel application to combinatorics. In the end, we connect our machinery to a recently introduced algorithm to locate eigenvalues of trees, and with our approach we show that any graph which contains two vertices separated by a unique path that is the subdivision of a bridge with at least six inner vertices cannot be integral. As a minor corollary this implies that most trees are not integral, but we believe no one thought otherwise.

##### 4.The distribution of the maximum protection number in simply generated trees

**Authors:**Clemens Heuberger, Sarah J. Selkirk, Stephan Wagner

**Abstract:** The protection number of a vertex $v$ in a tree is the length of the shortest path from $v$ to any leaf contained in the maximal subtree where $v$ is the root. In this paper, we determine the distribution of the maximum protection number of a vertex in simply generated trees, thereby refining a recent result of Devroye, Goh and Zhao. Two different cases can be observed: if the given family of trees allows vertices of outdegree $1$, then the maximum protection number is on average logarithmic in the tree size, with a discrete double-exponential limiting distribution. If no such vertices are allowed, the maximum protection number is doubly logarithmic in the tree size and concentrated on at most two values. These results are obtained by studying the singular behaviour of the generating functions of trees with bounded protection number. While a general distributional result by Prodinger and Wagner can be used in the first case, we prove a variant of that result in the second case.

##### 5.Permutations with few inversions

**Authors:**Anders Claesson, Atli Fannar Franklín, Einar Steingrímsson

**Abstract:** A curious generating function $S_0(x)$ for permutations of $[n]$ with exactly $n$ inversions is presented. Moreover, $(xC(x))^iS_0(x)$ is shown to be the generating function for permutations of $[n]$ with exactly $n-i$ inversions, where $C(x)$ is the generating function for the Catalan numbers.

##### 6.Infinite families of vertex-transitive graphs with prescribed Hamilton compression

**Authors:**Klavdija Kutnar, Dragan Marušič, Andriaherimanana Sarobidy Razafimahatratra

**Abstract:** Given a graph $X$ with a Hamilton cycle $C$, the {\em compression factor $\kappa(X,C)$ of $C$} is the order of the largest cyclic subgroup of $\operatorname{Aut}(C)\cap\operatorname{Aut}(X)$, and the {\em Hamilton compression $\kappa(X)$ of $X$ } is the maximum of $\kappa(X,C)$ where $C$ runs over all Hamilton cycles in $X$. Generalizing the well-known open problem regarding the existence of vertex-transitive graphs without Hamilton paths/cycles, it was asked by Gregor, Merino and M\"utze in [``The Hamilton compression of highly symmetric graphs'', {\em arXiv preprint} arXiv: 2205.08126v1 (2022)] whether for every positive integer $k$ there exists infinitely many vertex-transitive graphs (Cayley graphs) with Hamilton compression equal to $k$. Since an infinite family of Cayley graphs with Hamilton compression equal to $1$ was given there, the question is completely resolved in this paper in the case of Cayley graphs with a construction of Cayley graphs of semidirect products $\mathbb{Z}_p\rtimes\mathbb{Z}_k$ where $p$ is a prime and $k \geq 2$ a divisor of $p-1$. Further, infinite families of non-Cayley vertex-transitive graphs with Hamilton compression equal to $1$ are given. All of these graphs being metacirculants, some additional results on Hamilton compression of metacirculants of specific orders are also given.