A vertex coloring $ϕ$ of a graph $G$ is $p$-centered if for every connected subgraph $H$ of $G$ either $ϕ$ uses more than $p$ colors on $H$ or there is a color that appears exactly once on $H$. Centered colorings form one of the families of parameters that allow to capture notions of sparsity of graphs: A class of graphs has bounded expansion if and only if there is a function $f$ such that for every $p\geq1$, every graph in the class admits a $p$-centered coloring using at most $f(p)$ colors.
In this paper, we give upper bounds for the maximum number of colors needed in a $p$-centered coloring of graphs from several widely studied graph classes. We show that: (1) planar graphs admit $p$-centered colorings with $\mathcal{O}(p^3\log p)$ colors where the previous bound was $\mathcal{O}(p^{19})$; (2) bounded degree graphs admit $p$-centered colorings with $\mathcal{O}(p)$ colors while it was conjectured that they may require exponential number of colors in $p$; (3) graphs avoiding a fixed graph as a topological minor admit $p$-centered colorings with a polynomial in $p$ number of colors. All these upper bounds imply polynomial algorithms for computing the colorings. Prior to this work there were no non-trivial lower bounds known. We show that: (4) there are graphs of treewidth $t$ that require $\binom{p+t}{t}$ colors in any $p$-centered coloring and this bound matches the upper bound; (5) there are planar graphs that require $Ω(p^2\log p)$ colors in any $p$-centered coloring.
For $p\in \mathbb{N}$, a coloring $λ$ of the vertices of a graph $G$ is {\em{$p$-centered}} if for every connected subgraph~$H$ of $G$, either $H$ receives more than $p$ colors under $λ$ or there is a color that appears exactly once in $H$. In this paper, we prove that every $K_t$-minor-free graph admits a $p$-centered coloring with $\mathcal{O}(p^{g(t)})$ colors for some function $g$. In the special case that the graph is embeddable in a fixed surface $Σ$ we show that it admits a $p$-centered coloring with $\mathcal{O}(p^{19})$ colors, with the degree of the polynomial independent of the genu…
Square coloring is a variant of graph coloring where vertices within distance two must receive different colors. When considering planar graphs, the most famous conjecture (Wegner, 1977) states that $\frac32Δ+1$ colors are sufficient to square color every planar graph of maximum degree $Δ$. This conjecture has been proven asymptotically for graphs with large maximum degree. We consider here planar graphs with small maximum degree and show that $2Δ+7$ colors are sufficient, which improves the best known bounds when $6\leqslant Δ\leqslant 31$.
The Erdős-Gyárfás number $f(n, p, q)$ is the smallest number of colors needed to color the edges of the complete graph $K_n$ so that all of its $p$-clique spans at least $q$ colors. In this paper we improve the best known upper bound on $f(n, p, q)$ for many fixed values of $p, q$ and large $n$. Our proof uses a randomized coloring process, which we analyze using the so-called differential equation method to establish dynamic concentration.
Weak coloring numbers generalize the notion of degeneracy of a graph. They were introduced by Kierstead \& Yang in the context of games on graphs. Recently, several connections have been uncovered between weak coloring numbers and various parameters studied in graph minor theory and its generalizations. In this note, we show that for every fixed $k\geq1$, the maximum $r$-th weak coloring number of a graph with simple treewidth $k$ is $Θ(r^{k-1}\log r)$. As a corollary, we improve the lower bound on the maximum $r$-th weak coloring number of planar graphs from $Ω(r^2)$ to $Ω(r^2\log r)$, and we…
In this paper an improved bound on the chromatic number of the Pancake graph $P_n, n\geqslant 2$, is presented. The bound is obtained using a subadditivity property of the chromatic number of the Pancake graph. We also investigate an equitable coloring of $P_n$. An equitable $(n-1)$-coloring based on efficient dominating sets is given and optimal equitable $4$-colorings are considered for small $n$. It is conjectured that the chromatic number of $P_n$ coincides with its equitable chromatic number for any $n\geqslant 2$.
In this note, we improve on results of Hoppen, Kohayakawa and Lefmann about the maximum number of edge colorings without monochromatic copies of a star of a fixed size that a graph on $n$ vertices may admit. Our results rely on an improved application of an entropy inequality of Shearer.
Strengthening Hadwiger's conjecture, Gerards and Seymour conjectured in 1995 that every graph with no odd $K_t$-minor is properly $(t-1)$-colorable, this is known as the Odd Hadwiger's conjecture. We prove a relaxation of the above conjecture, namely we show that every graph with no odd $K_t$-minor admits a vertex $(2t-2)$-coloring such that all monochromatic components have size at most $\lceil \frac{1}{2}(t-2) \rceil$. The bound on the number of colors is optimal up to a factor of $2$, improves previous bounds for the same problem by Kawarabayashi (2008), Kang and Oum (2019), Liu and Wood (2…
The acyclic chromatic number of a graph is the least number of colors needed to properly color its vertices so that none of its cycles has only two colors. We show that for all $α>2^{-1/3}$ there exists an integer $Δ_α$ such that if the maximum degree $Δ$ of a graph is at least $Δ_α$, then the acyclic chromatic number of the graph is at most $\lceilαΔ^{4/3} \rceil +Δ+ 1$. The previous best bound, due to Gonçalves et al (2020), was $(3/2) Δ^{4/3} + O(Δ)$.
Given an undirected graph $G = (V,E)$, a conflict-free coloring with respect to open neighborhoods (CFON coloring) is a vertex coloring such that every vertex has a uniquely colored vertex in its open neighborhood. The minimum number of colors required for such a coloring is the CFON chromatic number of $G$, denoted by $χ_{ON}(G)$. In previous work [WG 2020], we showed the upper bound $χ_{ON}(G) \leq dc(G) + 3$, where $dc(G)$ denotes the distance to cluster parameter of $G$. In this paper, we obtain the improved upper bound of $χ_{ON}(G) \leq dc(G) + 1$. We also exhibit a family of graphs for …
A $(p,q)$-coloring of a graph $G$ is an edge-coloring of $G$ which assigns at least $q$ colors to each $p$-clique. The problem of determining the minimum number of colors, $f(n,p,q)$, needed to give a $(p,q)$-coloring of the complete graph $K_n$ is a natural generalization of the well-known problem of identifying the diagonal Ramsey numbers $r_k(p)$. The best-known general upper bound on $f(n,p,q)$ was given by Erdős and Gyárfás in 1997 using a probabilistic argument. Since then, improved bounds in the cases where $p=q$ have been obtained only for $p\in\{4,5\}$, each of which was proved by giv…
It is known that, for any $k$-list assignment $L$ of a graph $G$, the number of $L$-list colorings of $G$ is at least the number of the proper $k$-colorings of $G$ when $k>(m-1)/\ln(1+\sqrt{2})$. In this paper, we extend the Whitney's broken cycle theorem to $L$-colorings of signed graphs, by which we show that if $k> \binom{m}{3}+\binom{m}{4}+m-1$ then, for any $k$-assignment $L$, the number of $L$-colorings of a signed graph $Σ$ with $m$ edges is at least the number of the proper $k$-colorings of $Σ$. Further, if $L$ is $0$-free (resp., $0$-included) and $k$ is even (resp., odd), then the lo…
An odd coloring of a graph $G$ is a proper coloring such that any non-isolated vertex in $G$ has a coloring appears odd times on its neighbors. The odd chromatic number, denoted by $χ_o(G)$, is the minimum number of colors that admits an odd coloring of $G$. Petruševski and Škrekovski in 2021 introduced this notion and proved that if $G$ is planar, then $χ_o(G)\le9$ and conjectured that $χ_o(G)\le5$. More recently, Petr and Portier improved $9$ to $8$. A graph is $1$-planar if it can be drawn in the plane so that each edge is crossed by at most one other edge. Cranston, Lafferty and Song showe…
For a bipartite graph $G$ with parts $X$ and $Y$, an $X$-interval coloring is a proper edge coloring of $G$ by integers such that the colors on the edges incident to any vertex in $X$ form an interval. Denote by $χ'_{int}(G,X)$ the minimum $k$ such that $G$ has an $X$-interval coloring with $k$ colors. The author and Toft conjectured [Discrete Mathematics 339 (2016), 2628--2639] that there is a polynomial $P(x)$ such that if $G$ has maximum degree at most $Δ$, then $χ'_{int}(G,X) \leq P(Δ)$. In this short note, we prove this conjecture; in fact, we prove that a cubic polynomial suffices. We al…
We consider the coloring of certain distance graphs on the Euclidean plane. Namely, we ask for the minimal number of colors needed to color all points of the plane in such a way that pairs of points at distance in the interval $[1,b]$ get different colors. The classic Hadwiger-Nelson problem is a special case of this question -- obtained by taking $b=1$. The main results of the paper are improved lower and upper bounds on the number of colors for some values of $b$. In particular, we determine the minimal number of colors for two ranges of values of $b$ - one of which is enlarging an interval …
In an undirected graph $G$, a conflict-free coloring with respect to open neighborhoods (denoted by CFON coloring) is an assignment of colors to the vertices such that every vertex has a uniquely colored vertex in its open neighborhood. The minimum number of colors required for a CFON coloring of $G$ is the CFON chromatic number of $G$, denoted by $χ_{ON}(G)$. The decision problem that asks whether $χ_{ON}(G) \leq k$ is NP-complete. We obtain the following results: * Bodlaender, Kolay and Pieterse [WADS 2019] showed the upper bound $χ_{ON}(G)\leq {\sf fvs}(G)+3$, where ${\sf fvs}(G)$ denotes t…
The partition relation N \to (n)_{ \ell }^k means that whenever the k-tuples of an N-element set are \ell -colored, there is a monochromatic set of size n, where a set is called monochromatic if all its k-tuples have the same color. The logical negation of N \to (n)_{ \ell }^k is written as N \not \to (n)_{ \ell }^k. An ingenious construction of Erdős and Hajnal known as the stepping-up lemma gives a negative partition relation for higher uniformity from one of lower uniformity, effectively gaining an exponential in each application. Namely, if \ell \geq 2, k \geq 3, and N \not \to (n)_{ \ell …
A circular $r$-coloring of a signed graph $(G, σ)$ is an assignment $ϕ$ of points of a circle $C_r$ of circumference $r$ to the vertices of $(G, σ)$ such that for each positive edge $uv$ of $(G, σ)$ the distance of $ϕ(v)$ and $ϕ(v)$ is at least 1 and for each negative edge $uv$ the distance of $ϕ(u)$ from the antipodal of $ϕ(v)$ is at least 1. The circular chromatic number of $(G, σ)$, denoted $χ_c(G, σ)$, is the infimum of $r$ such that $(G, σ)$ admits a circular $r$-coloring. This notion is recently defined by Naserasr, Wang, and Zhu who, among other results, proved that for any signed $d$-d…
Université Clermont Auvergne, Clermont Auvergne INP（克莱蒙奥弗涅大学）；FAMNIT and IAM, University of Primorska（普里莫斯卡大学 FAMNIT 和 IAM）；CNRS, ENS de Lyon, Université Claude Bernard Lyon 1（法国国家科学研究中心，里昂高等师范学院，克洛德·贝尔纳里昂第一大学）
An edge-coloring of a graph $G$ with colors $1,\ldots,t$ is called an \emph{interval $t$-coloring} if all colors are used and the colors of edges incident to each vertex of $G$ are distinct and form an interval of integers. In 1990, Kamalian proved that if a graph $G$ with at least one edge has an interval $t$-coloring, then $t\leq 2|V(G)|-3$. In 2002, Axenovich improved this upper bound for planar graphs: if a planar graph $G$ admits an interval $t$-coloring, then $t\leq \frac{11}{6}|V(G)|$. In the same paper Axenovich suggested a conjecture that if a planar graph $G$ has an interval $t$-colo…
A ladder is a $2 \times k$ grid graph. When does a graph class $\mathcal{C}$ exclude some ladder as a minor? We show that this is the case if and only if all graphs $G$ in $\mathcal{C}$ admit a proper vertex coloring with a bounded number of colors such that for every $2$-connected subgraph $H$ of $G$, there is a color that appears exactly once in $H$. This type of vertex coloring is a relaxation of the notion of centered coloring, where for every connected subgraph $H$ of $G$, there must be a color that appears exactly once in $H$. The minimum number of colors in a centered coloring of $G$ is…
Much recent progress in hypergraph Ramsey theory has focused on constructions that lead to lower bounds for the corresponding Ramsey numbers. In this paper, we consider applications of these results to Gallai colorings. That is, we focus on the Ramsey numbers resulting from only considering $t$-colorings of the hyperedges of complete $r$-uniform hypergraphs in which no rainbow $K_{r+1}^{(r)}$-subhypergraphs exists. We also provide new constructions which imply improved lower bounds for many $3$ and $4$-uniform Ramsey numbers and $3$ and $4$-uniform Gallai-Ramsey numbers.
The multicolor Ramsey number problem asks, for each pair of natural numbers $\ell$ and $t$, for the largest $\ell$-coloring of a complete graph with no monochromatic clique of size $t$. Recent works of Conlon-Ferber and Wigderson have improved the longstanding lower bound for this problem. We make a further improvement by replacing an explicit graph appearing in their constructions by a random graph. Graphs useful for this construction are exactly those relevant for a problem of Erdős on graphs with no large cliques and few large independent sets. We also make some basic observations about thi…
For a given graph $H$, the Ramsey number $r(H)$ is the minimum $N$ such that any 2-edge-coloring of the complete graph $K_N$ yields a monochromatic copy of $H$. Given a positive integer $n$, a \emph{fan }$F_n$ is a graph formed by $n$ triangles that share one common vertex. We show that ${9n}/{2}-5\le r(F_n)\le {11n}/{2} + 6$ for any $n$. This improves previous best bounds $r(F_n) \le 6n$ of Lin and Li and $r(F_n) \ge 4n+2$ of Zhang, Broersma and Chen.
