Combinatorics (math.CO)

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 $r$.

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.

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.