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Kostochka and Woodall (2001) conjectured that the square of every graph has the same chromatic number and list chromatic number. In 2015 Kim and Park disproved this conjecture for non-bipartite and bipartite graphs. It was asked by several…

The bivariate chromatic polynomial $\chi_G(x,y)$ of a graph $G = (V, E)$, introduced by Dohmen-P\"{o}nitz-Tittmann (2003), counts all $x$-colorings of $G$ such that adjacent vertices get different colors if they are $\le y$. We extend this…

For a multigraph $G$, $\chi'(G)$ denotes the chromatic index of $G$, $\Delta(G)$ the maximum degree of $G$, and $\Gamma(G) = \max\left\{\left\lceil \frac{2|E(H)|}{|V(H)|-1} \right\rceil: H \subseteq G \text{ and } |V(H)| \text{…

Let $\text{ch}(G)$ denote the choice number of a graph $G$ (also called "list chromatic number" or "choosability" of $G$). Noel, Reed, and Wu proved the conjecture of Ohba that $\text{ch}(G)=\chi(G)$ when $|V(G)|\le 2\chi(G)+1$. We extend…

The chromatic number $\chi(G)$ of a graph $G$, that is, the smallest number of colors required to color the vertices of $G$ so that no two adjacent vertices are assigned the same color, is a classic and extensively studied parameter. Here…

Let $G$ be a graph with chromatic number $\chi$, maximum degree $\Delta$ and clique number $\omega$. Reed's conjecture states that $\chi \leq \lceil (1-\varepsilon)(\Delta + 1) + \varepsilon\omega \rceil$ for all $\varepsilon \leq 1/2$. It…

The Hom-complexes were introduced by Lovasz to study topological obstructions to graph colorings. It was conjectured by Babson and Kozlov, and proved by Cukic and Kozlov, that Hom(G,K_n) is (n-d-2)-connected, where d is the maximal degree…

An edge coloring of a graph $G$ is called conflict-free if, for every edge, its closed neighborhood contains a color that appears exactly once. The least number of colors required for such a coloring is the conflict-free chromatic index of…

A class of graphs $\mathcal{G}$ is $\chi$-bounded if there exists a function $f$ such that $\chi(G) \leq f(\omega(G))$ for each graph $G \in \mathcal{G}$, where $\chi(G)$ and $\omega(G)$ are the chromatic and clique number of $G$,…

Given a graph $G$, a vertex-colouring $\sigma$ of $G$, and a subset $X\subseteq V(G)$, a colour $x \in \sigma(X)$ is said to be \emph{odd} for $X$ in $\sigma$ if it has an odd number of occurrences in $X$. We say that $\sigma$ is an…

The distinguishing chromatic number, $\chi_D(G)$, of a graph $G$ is the smallest number of colors in a proper coloring, $\varphi$, of $G$, such that the only automorphism of $G$ that preserves all colors of $\varphi$ is the identity map.…

The closed neighborhood conflict-free chromatic number of a graph $G$, denoted by $\chi_{CN}(G)$, is the minimum number of colors required to color the vertices of $G$ such that for every vertex, there is a color that appears exactly once…

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…

The Erd\H{o}s-Lov\'asz Tihany Conjecture states that any $G$ with chromatic number $\chi(G) = s + t - 1 > \omega(G)$, with $s,t \geq 2$ can be split into two vertex-disjoint subgraphs of chromatic number $s, t$ respectively. We prove this…

Given a graph $H$, let us denote by $f_\chi(H)$ and $f_\ell(H)$, respectively, the maximum chromatic number and the maximum list chromatic number of $H$-minor-free graphs. Hadwiger's famous coloring conjecture from 1943 states that…

Several games that arise from graph coloring have been introduced and studied. Let $\varphi$ denote a graph invariant that arises from such a game. If $G$ is a graph and $\varphi(G-x)\neq \varphi(G)=k$, $k \geq 1$, holds true for every…

The distinguishing chromatic number of a graph $G$, denoted $\chi_D(G)$, is the minimum number of colours in a proper vertex colouring of $G$ that is preserved by the identity automorphism only. Collins and Trenk proved that $\chi_D(G)\le…

A dynamic coloring of a graph $G$ is a proper coloring such that for every vertex $v\in V(G)$ of degree at least 2, the neighbors of $v$ receive at least 2 colors. It was conjectured [B. Montgomery. {\em Dynamic coloring of graphs}. PhD…

In the 70s, Goldberg, and independently Seymour, conjectured that for any multigraph $G$, the chromatic index $\chi'(G)$ satisfies $\chi'(G)\leq \max \{\Delta(G)+1, \lceil\rho(G)\rceil\}$, where $\rho(G)=\max \{\frac {e(G[S])}{\lfloor…

The \emph{choice number} of a graph $G$, denoted $\ch(G)$, is the minimum integer $k$ such that for any assignment of lists of size $k$ to the vertices of $G$, there is a proper colouring of $G$ such that every vertex is mapped to a colour…