Let $G$ be a quadrangulation on an orientable surface and let $g$ be a proper vertex-$4$-coloring of $G$. A face $F$ of $G$ is said to be a rainbow-face if all four distinct colors appear on its boundary. A $(c_1,c_2,c_3,c_4)$-face in $G$ is a rainbow face with colors $c_i$, $i=1,2,3,4$ on the boundary in clockwise order. We show that the number of $(c_1,c_2,c_3,c_4)$-faces in $G$ equals the number of $(c_4,c_3,c_2,c_1)$-faces. This implies in particular that the number of rainbow-faces of $G$ is even.
We give a linear-time algorithm to decide 3-colorability (and find a 3-coloring, if it exists) of quadrangulations of a fixed surface. The algorithm also allows to prescribe the coloring for a bounded number of vertices.
A graph drawn in a surface is a near-quadrangulation if the sum of the lengths of the faces different from 4-faces is bounded by a fixed constant. We leverage duality between colorings and flows to design an efficient algorithm for 3-precoloring-extension in near-quadrangulations of orientable surfaces. Furthermore, we use this duality to strengthen previously known sufficient conditions for 3-colorability of triangle-free graphs drawn in orientable surfaces.
Sperner's lemma states that every Sperner coloring of a triangulation of a simplex contains a fully colored simplex. We present a generalization of this lemma, where instead of triangulations are considered quadrangulations.
Let G be a simple connected plane graph and let C_1 and C_2 be cycles in G bounding distinct faces f_1 and f_2. For a positive integer l, let r(l) denote the number of integers n such that -l<=n<=l, n is divisible by 3, and n has the same parity as l; in particular, r(4)=1. Let r_{f_1,f_2}(G) be the product of r(|f|) over all faces f of G distinct from f_1 and f_2, and let q(G)=1+sum_{f:|f|\neq 4} |f|, where the sum is over all faces f of G. We give an algorithm with time complexity O(r_{f_1,f_2}(G)q(G)|G|) which, given a 3-coloring psi of C_1 and C_2, either finds an extension of psi to a 3-c…
Higher-dimensional generalization of Youngs' theorem and circular colorings
College of Liberal Arts and Science, Kitasato University（北里大学文理学院）；Department of Mathematical Sciences, Faculty of Science, Shinshu University（信州大学理学部数学科学系）
A facial unique-maximum coloring of a plane graph is a vertex coloring where on each face $α$ the maximal color appears exactly once on the vertices of $α$. If the coloring is required to be proper, then the upper bound for the minimal number of colors required for such a coloring is set to $5$. Fabrici and Göring [Fabrici and Goring 2016] even conjectured that $4$ colors always suffice. Confirming the conjecture would hence give a considerable strengthening of the Four Color Theorem. In this paper, we prove that the conjecture holds for subcubic plane graphs, outerplane graphs and plane quadr…
Université Paris Cité（巴黎西岱大学）；CNRS（法国国家科学研究中心）；IRIF（信息与随机研究所）；Zhejiang Normal University（浙江师范大学）；Hainan Bielefeld University of Applied Sciences（海南比勒费尔德应用科技大学）
A fall coloring of a graph G is a proper coloring of G with k colors such that each vertex sees all k colors on its closed neighborhood. In this short note, we characterize all fall colorings of Kneser graphs of type KG(n,2).
Let $t\geqslant 2$ and $s\geqslant 1$ be two integers. Define a $(t,s)$-coloring of a hypergraph to be a coloring of its vertices using $t$ colors such that each color appears on each edge at least $s$ times. In this note, we provide a sufficient condition for the existence of a $(t,s)$-coloring of a hypergraph by using the symmetric lopsided version of Lovász Local Lemma. Our result generalizes several known results on hypergraph colorings.
Dvořák and Postle \cite{DP} introduced a \textit{DP-coloring} of a simple graph as a generalization of a list-coloring. They proved a Brooks' type theorem for a DP-coloring, and Bernshteyn, Kostochka and Pron \cite{BKP} extended it to a DP-coloring of multigraphs. However, detailed structure when a multigraph does not admit a DP-coloring was not specified in \cite{BKP}. In this note, we make this point clear and give the complete structure. This is also motivated by the relation to signed coloring of signed graphs.
In this short note, the purpose is to provide an upper bound for the b-chromatic number of Kneser graphs. Our bound improves the upper bound that was presented by Balakrishnan and Kavaskar in [b-coloring of Kneser graphs, Discrete Appl. Math. 160 (2012), 9-14].
A balanced edge-coloring of the complete graph is an edge-coloring such that every vertex is incident to each color the same number of times. In this short note, we present a construction of a balanced edge-coloring with six colors of the complete graph on $n=13^k$ vertices, for every positive integer $k$, with no rainbow $K_4$. This solves a problem by Erdős and Tuza.
{\emph A star edge-coloring} of a graph is a proper edge-coloring without bichromatic paths and cycles of length four. In this paper, we consider the list version of this coloring and prove that the list star chromatic index of every subcubic graph is at most $7$, answering the question of Dvořák et al. (Star chromatic index, J. Graph Theory 72 (2013), 313--326).
We prove that it is always possible to color online nonrepetitively any (partial) $k$-tree (that is, graphs with tree-width at most $k$) with $4^k$ colors. This implies that it is always possible to color online nonrepetitively cycles, trees and series-parallel graphs with $16$ colors. Our results generalize the respective (offline) nonrepetitive coloring results.
We consider the \emph{connected game coloring number} of a graph, introduced by Charpentier et al. as a game theoretic graph parameter that measures the degeneracy of a graph with respect to a certain two-player game played with an uncooperative adversary. We consider the connected game coloring number of graphs of bounded treedepth and of $k$-trees. In particular, we show that there exists an outerplanar $2$-tree with connected game coloring number of $5$, which answers a question from [C. Charpentier, H. Hocquard, E. Sopena, and X. Zhu. A connected version of the graph coloring game. \textit…
We prove that the acyclic chromatic number of a graph with maximum degree $Δ$ is less than $2.835Δ^{4/3}+Δ$. This improves the previous upper bound, which was $50Δ^{4/3}$. To do so, we draw inspiration from works by Alon, McDiarmid and Reed and by Esperet and Parreau.
Even-hole-free graphs are a graph class of much interest. Foley et al. [Graphs Comb. 36(1): 125-138 (2020)] have recently studied $(4K_1, C_4, C_6)$-free graphs, which form a subclass of even-hole-free graphs. Specifically, Foley et al. have shown an algorithm for coloring these graphs via bounded clique-width if they contain a $C_7$. In this note, we give a simpler and much faster algorithm via a more restrictive graph parameter, neighborhood diversity.
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…
Let $G$ be a simple graph with $n$ vertices. The coloring complex $Δ(G)$ was defined by Steingr\'ımsson, and the homology of $Δ(G)$ was shown to be nonzero only in dimension $n-3$ by Jonsson. Hanlon recently showed that the Eulerian idempotents provide a decomposition of the homology group $H_{n-3}(Δ(G))$ where the dimension of the $j^{th}$ component in the decomposition, $H_{n-3}^{(j)}(Δ(G))$, equals the absolute value of the coefficient of $λ^{j}$ in the chromatic polynomial of $G$, $χ_{G}(λ)$. Jonsson recently studied the topology of intersections of coloring complexes. In this note, we sho…
A linear hypergraph is intersecting if any two different edges have exactly one common vertex and an $n$-quasicluster is an intersecting linear hypergraph with $n$ edges each one containing at most $n$ vertices and every vertex is contained in at least two edges. The Erdös-Faber-Lovász Conjecture states that the chromatic number of any $n$-quasicluster is at most $n$. In the present note we prove the correctness of the conjecture for a new infinite class of $n$-quasiclusters using a specific edge coloring of the complete graph.
A connected simple graph is said dual-hamiltonian if its vertex set has a $2$-coloring such that each color class induces a tree. We call such a coloring a hamiltonian coloring. We prove that if $G$ is a graph with a certain type of hamiltonian coloring and $T$ is a tree, then $G\times T$ is also dual-hamiltonian having the same certain type of hamiltonian coloring. This result is used to constructed a class of dual-hamiltonian graphs, which includes the hypercubes and other multidimensional grids.
We prove bounds on the chromatic number $χ$ of a vertex-transitive graph in terms of its clique number $ω$ and maximum degree $Δ$. We conjecture that every vertex-transitive graph satisfies $χ\le \max \left\{ω, \left\lceil\frac{5Δ+ 3}{6}\right\rceil\right\}$ and we prove results supporting this conjecture. Finally, for vertex-transitive graphs with $Δ\ge 13$ we prove the Borodin-Kostochka conjecture, i.e., $χ\le\max\{ω,Δ-1\}$.
A graph is locally irregular if the degrees of the end-vertices of every edge are distinct. An edge coloring of a graph G is locally irregular if every color induces a locally irregular subgraph of G. A colorable graph G is any graph which admits a locally irregular edge coloring. The locally irregular chromatic index X'irr(G) of a colorable graph G is the smallest number of colors required by a locally irregular edge coloring of G. The Local Irregularity Conjecture claims that all colorable graphs require at most 3 colors for a locally irregular edge coloring. Recently, it has been observed t…
A Gallai coloring of a complete graph is an edge-coloring such that no triangle has all its edges colored differently. A Gallai $k$-coloring is a Gallai coloring that uses $k$ colors. Given a graph $H$ and an integer $k\geq 1$, the Gallai-Ramsey number $GR_k(H)$ of $H$ is the least positive integer $N$ such that every Gallai $k$-coloring of the complete graph $K_N$ contains a monochromatic copy of $H$. Let $W_{2n} $ denote an even wheel on $2n+1\ge5$ vertices. In this note, we study Gallai-Ramsey number of $W_{2n}$ and completely determine the exact value of $GR_k(W_4)$ for all $k\ge2$.
Recently, Kim and Park have found an infinite family of graphs whose squares are not chromatic-choosable. Xuding Zhu asked whether there is some $k$ such that all $k$th power graphs are chromatic-choosable. We answer this question in the negative: we show that there is a positive constant $c$ such that for any $k$ there is a family of graphs $G$ with $χ(G^k)$ unbounded and $χ_{\ell}(G^k)\geq c χ(G^k) \log χ(G^k)$. We also provide an upper bound, $χ_{\ell}(G^k)<χ(G^k)^3$ for $k>1$.
For two graphs, $G$, and $H$, an edge-coloring of a complete graph is $(G,H)$-good if there is no monochromatic subgraph isomorphic to $G$ and no rainbow subgraph isomorphic to $H$ in this coloring. The set of number of colors used by some $(G,H)$-colorings of $K_n$ is called a mixed-Ramsey spectrum. This note addresses a fundamental question of whether the spectrum is an interval. It is shown that the answer is "yes" if $G$ is not a star and $H$ does not contain a pendent edge.
This paper studies the quantity $p(n,r)$, that is the minimal number of edges of an $n$-uniform hypergraph without panchromatic coloring (it means that every edge meets every color) in $r$ colors. If $r \leq c \frac{n}{\ln n}$ then all bounds have a type $A_1(n, \ln n, r)(\frac{r}{r-1})^n \leq p(n, r) \leq A_2(n, r, \ln r) (\frac{r}{r-1})^n$, where $A_1$, $A_2$ are some algebraic fractions. The main result is a new lower bound on $p(n,r)$ when $r$ is at least $c \sqrt n$; we improve an upper bound on $p(n,r)$ if $n = o(r^{3/2})$. Also we show that $p(n,r)$ has upper and lower bounds depend onl…
The digirth of a digraph is the length of a shortest directed cycle. The dichromatic number $\vecχ(D)$ of a digraph $D$ is the smallest size of a partition of the vertex-set into subsets inducing acyclic subgraphs. A conjecture by Harutyunyan and Mohar states that $\vecχ(D) \le \left\lceil\fracΔ{4}\right\rceil+1$ for every digraph $D$ of digirth at least $3$ and maximum degree $Δ$. The best known partial result by Golowich shows that $\vecχ(D) \le \frac{2}{5}Δ+O(1)$. In this short note we prove for every $g \ge 2$ that if $D$ is a digraph of digirth at least $2g-1$ and maximum degree $Δ$, then…
In this note, we prove that the base case of the Graham--Rothschild Theorem, i.e., the one that considers colorings of the ($1$-dimensional) variable words, admits bounds in the class $\mathcal{E}^5$ of Grzegorczyk's hierarchy.
In 1973, Altshuler characterized the $6$-regular triangulations on the torus to be precisely those that are obtained from a regular triangulation of the $r \times s$ toroidal grid where the vertices in the first and last column are connected by a shift of $t$ vertices. Such a graph is denoted $T(r, s, t)$. In 1999, Collins and Hutchinson classified the $4$-colorable graphs $T(r, s, t)$ with $r, s \geq 3$. In this paper, we point out a gap in their classification and show how it can be fixed. Combined with the classification of the $4$-colorable graphs $T(1, s, t)$ by Yeh and Zhu in 2003, this …
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.
DP-coloring (also called correspondence coloring) is a generalization of list coloring recently introduced by Dvořák and Postle. Several known bounds for the list chromatic number of a graph $G$, $χ_\ell(G)$, also hold for the DP-chromatic number of $G$, $χ_{DP}(G)$. On the other hand, there are several properties of the DP-chromatic number that shows that it differs with the list chromatic number. In this note we show one such property. It is well known that $χ_\ell (K_{k,t}) = k+1$ if and only if $t \geq k^k$. We show that $χ_{DP} (K_{k,t}) = k+1$ if $t \geq 1 + (k^k/k!)(\log(k!)+1)$, and we…
The Grundy number of a graph $G$ is the maximum number of colors used by the First-Fit coloring of $G$ and is denoted by $Γ(G)$. Similarly, the ${\rm b}$-chromatic number ${\rm{b}}(G)$ of $G$ expresses the worst case behavior of another well-known coloring procedure i.e. color-dominating coloring of $G$. We obtain some families of graphs $\mathcal{F}$ for which there exists a function $f(x)$ such that $Γ(G)\leq f({\rm{b}}(G))$, for each graph $G$ from the family. Call any such family $(Γ,b)$-bounded family. We conjecture that the family of ${\rm b}$-monotone graphs is $(Γ,b)$-bounded and valid…
DP-coloring (also known as correspondence coloring) is a generalization of list coloring introduced recently by Dvořák and Postle. Many known upper bounds for the list-chromatic number extend to the DP-chromatic number, but not all of them do. In this note we describe some unusual properties of DP-coloring that set it aside from list coloring. In particular, we give an example of a planar bipartite graph with DP-chromatic number $4$ and prove that the edge-DP-chromatic number of a $d$-regular graph with $d\geq 2$ is always at least $d+1$.
A star edge coloring of a graph $G$ is a proper edge coloring of $G$ without bichromatic paths or cycles of length four. The it star chromatic index, $χ_{st}^{'} (G ),$ of $G$ is the minimum number $k$ for which $G$ has a star edge coloring by $k$ colors. In \cite{LB}, L. Bezegov$\acute{a}$ et al. conjectured that $χ_{st}^{'} (G )\leq \lfloor\frac{3Δ}{2}\rfloor+1$ when $G$ is an outerplanar graph with maximum degree $Δ\geq 3.$ In this paper we obtained that $χ_{st}^{'}(G) \leq Δ+6$ when $G$ is an 2-connected outerplanar graph with diameter 2 or 3. If $G$ is an 2-connected outerplanar graph wit…
A tree $T$, in an edge-colored graph $G$, is called {\em a rainbow tree} if no two edges of $T$ are assigned the same color. For a vertex subset $S\in V(G)$, a tree that connects $S$ in $G$ is called an $S$-tree. A {\em $k$-rainbow coloring} of $G$ is an edge coloring of $G$ having the property that for every set $S$ of $k$ vertices of $G$, there exists a rainbow $S$-tree $T$ in $G$. The minimum number of colors needed in a $k$-rainbow coloring of $G$ is the {\em $k$-rainbow index of $G$}, denoted by $rx_k(G)$. In this paper, we obtain the exact values of $rx_3(K_{2,t})$ for any $t\geq 1$.
