Let $G=(V,E)$ be a graph. A (proper) $k$-edge-coloring is a coloring of the edges of $G$ such that any pair of edges sharing an endpoint receive distinct colors. A classical result of Vizing ensures that any simple graph $G$ admits a $(Δ(G)+1)$-edge coloring where $Δ(G)$ denotes the maximum degreee of $G$. Recently, Cabello raised the following question: given two graphs $G_1,G_2$ of maximum degree $Δ$ on the same set of vertices $V$, is it possible to edge-color their (edge) union with $Δ+2$ colors in such a way the restriction of $G$ to respectively the edges of $G_1$ and the edges of $G_2$ are edge-colorings? More generally, given $\ell$ graphs, how many colors do we need to color their union in such a way the restriction of the coloring to each graph is proper?
In this short note, we prove that we can always color the union of the graphs $G_1,\ldots,G_\ell$ of maximum degree $Δ$ with $Ω(\sqrt{\ell} \cdot Δ)$ colors and that there exist graphs for which this bound is tight up to a constant multiplicative factor. Moreover, for two graphs, we prove that at most $\frac 32 Δ+4$ colors are enough which is, as far as we know, the best known upper bound.
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.
The smallest number of edges forming an n-uniform hypergraph which is not r-colorable is denoted by m(n,r). Erdős and Lovász conjectured that m(n,2)=θ(n 2^n)$. The best known lower bound m(n,2)=Ω(sqrt(n/log(n)) 2^n) was obtained by Radhakrishnan and Srinivasan in 2000. We present a simple proof of their result. The proof is based on analysis of random greedy coloring algorithm investigated by Pluhár in 2009. The proof method extends to the case of r-coloring, and we show that for any fixed r we have m(n,r)=Ω((n/log(n))^(1-1/r) r^n) improving the bound of Kostochka from 2004. We also derive ana…
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 proper edge coloring of a simple graph $G$ is called a vertex distinguishing edge coloring (vdec) if for any two distinct vertices $u$ and $v$ of $G$, the set of the colors assigned to the edges incident to $u$ differs from the set of the colors assigned to the edges incident to $v$. The minimum number of colors required for all vdecs of $G$ is denoted by $χ\,'_s(G)$ called the vdec chromatic number of $G$. Let $n_d(G)$ denote the number of vertices of degree $d$ in $G$. In this note, we show that a tree $T$ with $n_2(T)\leq n_1(T)$ holds $χ\,'_s(T)=n_1(T)+1$ if its diameter $D(T)=3$ or one …
A {\em hole} is a chordless cycle of length at least four. A hole is {\em even} (resp. {\em odd}) if it contains an even (resp. odd) number of vertices. A \emph{cap} is a graph induced by a hole with an additional vertex that is adjacent to exactly two adjacent vertices on the hole. In this note, we use a decomposition theorem by Conforti et al. (1999) to show that if a graph $G$ does not contain any even hole or cap as an induced subgraph, then $χ(G)\le \lfloor\frac{3}{2}ω(G)\rfloor$, where $χ(G)$ and $ω(G)$ are the chromatic number and the clique number of $G$, respectively. This bound is at…
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.
In Raman (2007), using a column construction technique it is proved that every interval graph can be colored online with First Fit with at most $8w(G)$ colors, where $w(G)$ is the size of the maximum clique of $G$. Since the column construction can not be adapted to circular arc graphs we give a different proof to establish an upper bound of $9w(G)$ for online coloring a circular arc graph $G$ with the First Fit algorithm.
A proof that every outerplanar graph is Δ+2 colorable. This is slightly stronger then an unpublished result of Wang Shudong, Ma Fangfang, Xu Jin, and Yan Lijun proving the same for 2-connected outerplanar graphs.
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 present a simpler proof of a bound on the number of proper colorings of a graph that was obtained recently by Liu and Murty using Tur'an sieve (in fact, we prove a stronger inequality). We also point out that these results are subsumed in a stronger result due to Lazebnik in 1990.
A $vi$-simultaneous proper $k$-coloring of a graph $G$ is a coloring of all vertices and incidences of the graph in which any two adjacent or incident elements in the set $V(G)\cup I(G)$ receive distinct colors, where $I(G)$ is the set of incidences of $G$. The $vi$-simultaneous chromatic number, denoted by $χ_{vi}(G)$, is the smallest integer $k$ such that $G$ has a $vi$-simultaneous proper $k$-coloring. In [M. Mozafari-Nia, M. N. Iradmusa, A note on coloring of $\frac{3}{3}$-power of subquartic graphs, Vol. 79, No.3, 2021] $vi$-simultaneous proper coloring of graphs with maximum degree $4$ i…
Here we observe that list coloring in graph theory coincides with the zero-temperature antiferromagnetic Potts model with an external field. We give a list coloring polynomial that equals the partition function in this case. This is analogous to the well-known connection between the chromatic polynomial and the zero-temperature, zero-field, antiferromagnetic Potts model. The subsequent cross fertilization yields immediate results for the Potts model and suggests new research directions in list coloring.
A packing $k$-coloring for some integer $k$ of a graph $G=(V,E)$ is a mapping $φ:V\to\{1,\ldots,k\}$ such that any two vertices $u, v$ of color $φ(u)=φ(v)$ are in distance at least $φ(u)+1$. This concept is motivated by frequency assignment problems. The \emph{packing chromatic number} of $G$ is the smallest $k$ such that there exists a packing $k$-coloring of $G$. Fiala and Golovach showed that determining the packing chromatic number for chordal graphs is \NP-complete for diameter exactly 5. While the problem is easy to solve for diameter 2, we show \NP-completeness for any diameter at least…
In this note we prove that K_{1,m,n} is interval edge-colorable if and only if gcd(m+1,n+1)=1. It settles in the affirmative a conjecture of Petrosyan.
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., $χ\le\max\{ω,Δ-1\}$.
Let $G$ be a simple graph with $n$ vertices and list chromatic number $χ_\ell(G)=χ_\ell$. Suppose that $0\leq t\leq χ_\ell$ and each vertex of $G$ is assigned a list of $t$ colors. Albertson, Grossman and Haas [1] conjectured that at least $\frac{tn}{χ_\ell}$ vertices of $G$ can be colored from these lists. In this paper we find some new results in partial list coloring which help us to show that the conjecture is true for at least half of the numbers of the set $\{1,2,...,χ_\ell(G)-1\}$. In addition we introduce a new related conjecture and finally we present some results about this conjectur…
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…
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.
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.
We show that the lines of every arrangement of $n$ lines in the plane can be colored with $O(\sqrt{n/ \log n})$ colors such that no face of the arrangement is monochromatic. This improves a bound of Bose et al. \cite{BCC12} by a $Θ(\sqrt{\log n})$ factor. Any further improvement on this bound will improve the best known lower bound on the following problem of Erdős: Estimate the maximum number of points in general position within a set of $n$ points containing no four collinear points.
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 a proper edge-coloring of a cubic graph an edge $uv$ is called poor or rich, if the set of colors of the edges incident to $u$ and $v$ contains exactly three or five colors, respectively. An edge-coloring of a graph is normal, if any edge of the graph is either poor or rich. In this note, we show that some snarks constructed by using a method introduced by Loupekhine admit a normal edge-coloring with five colors. The existence of a Berge-Fulkerson Covering for a part of the snarks considered in this paper was recently proved by Manuel and Shanthi (2015). Since the existence of a normal edge…
A signed tree-coloring of a signed graph $(G,σ)$ is a vertex coloring $c$ so that $G^{c}(i,\pm)$ is a forest for every $i\in c(u)$ and $u\in V(G)$, where $G^{c}(i,\pm)$ is the subgraph of $(G,σ)$ whose vertex set is the set of vertices colored by $i$ or $-i$ and edge set is the set of positive edges with two end-vertices colored both by $i$ or both by $-i$, along with the set of negative edges with one end-vertex colored by $i$ and the other colored by $-i$. If $c$ is a function from $V(G)$ to $M_n$, where $M_n$ is $\{\pm 1,\pm 2,\ldots,\pm k\}$ if $n=2k$, and $\{0,\pm 1,\pm 2,\ldots,\pm k\}$ …
