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Combinatorics (math.CO)

Thu, 25 May 2023

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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.