Combinatorics (math.CO)

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.

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.

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.

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 $D \equiv 0 \pmod{6}$.

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.

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.

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 $\Delta(G) \ge |V(G)|-3$.

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.

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.

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 $(x-x')^2 + (y-y')^2 - (z-z')^2 = 1$.