Related papers: List Coloring a Cartesian Product with a Complete …

Let $G$ be a graph and $t$ a nonnegative integer. Suppose $f$ is a mapping from the vertex set of $G$ to $\{1,2,\dots, k\}$. If, for any vertex $u$ of $G$, the number of neighbors $v$ of $u$ with $f(v)=f(u)$ is less than or equal to $t$,…

A fall $k$-coloring of a graph $G$ is a proper $k$-coloring of $G$ such that each vertex of $G$ sees all $k$ colors on its closed neighborhood. We denote ${\rm Fall}(G)$ the set of all positive integers $k$ for which $G$ has a fall…

For a positive integer $k$, a proper $k$-coloring of a graph $G$ is a mapping $f: V(G) \rightarrow \{1,2, \ldots, k\}$ such that $f(u) \neq f(v)$ for each edge $uv$ of $G$. The smallest integer $k$ for which there is a proper $k$-coloring…

Recently, Chen and Deng have proved that every graph $G$ with independence number two contains $K_{\ell,\chi(G)- \ell}$ as a minor for each integer $\ell$ with $1\leq\ell < \chi(G)$. In this paper, we extend this result to odd minor…

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 Hadwiger number mr(G) of a graph G is the largest integer n for which the complete graph K_n on n vertices is a minor of G. Hadwiger conjectured that for every graph G, mr(G) >= chi(G), where chi(G) is the chromatic number of G. In this…

A proper vertex coloring of a graph is equitable if the sizes of color classes differ by at most 1. The equitable chromatic number of a graph $G$, denoted by $\chi_=(G)$, is the minimum $k$ such that $G$ is equitably $k$-colorable. The…

Kr\'al' and Sgall (2005) introduced a refinement of list colouring where every colour list must be subset to one predetermined palette of colours. We call this $(k,\ell)$-choosability when the palette is of size at most $\ell$ and the lists…

DP-coloring (also known as correspondence coloring) is a generalization of list coloring developed recently by Dvo\v{r}\'{a}k and Postle. In this paper we introduce and study the fractional DP-chromatic number $\chi_{DP}^\ast(G)$. We…

In this paper, we investigate the following Gallai-Ramsey question: how large must a complete bipartite graph $K_{n_1, n_2}$ be before any coloring of its edges with $r$ colors contains either a monochromatic copy of $G = K_{s,t}$ or a…

A $b$-coloring of a graph $G$ is a proper coloring of its vertices such that each color class contains a vertex that has at least one neighbor in all the other color classes. The b-Coloring problem asks whether a graph $G$ has a…

A graph $G$ is $r$-equitably $k$-colorable if its vertex set can be partitioned into $k$ independent sets, any two of which differ in size by at most $r$. The $r$-equitable chromatic threshold of a graph $G$, denoted by $\chi_{r=}^*(G)$, is…

A \emph{domatic} (\emph{total domatic}) \emph{$k$-coloring} of a graph $G$ is an assignment of $k$ colors to the vertices of $G$ such that each vertex contains vertices of all $k$ colors in its closed neighborhood (neighborhood). The…

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…

A decomposition of a non-empty simple graph $G$ is a pair $[G,P]$, such that $P$ is a set of non-empty induced subgraphs of $G$, and every edge of $G$ belongs to exactly one subgraph in $P$. The chromatic index $\chi'([G,P])$ of a…

Borodin and Kostochka in 1977 conjectured that if a graph $G$ has maximum degree $\Delta(G)\ge 9$ and its clique number satisfies $\omega(G)\le \Delta(G)-1$, then its chromatic number satisfies $\chi(G) \le \Delta(G)-1$. We prove this…

A proper vertex $k$-coloring of a graph $G=(V,E)$ is an assignment $c:V\to \{1,2,\ldots,k\}$ of colors to the vertices of the graph such that no two adjacent vertices are associated with the same color. The square $G^2$ of a graph $G$ is…

For any graph $G$, the First-Fit (or Grundy) chromatic number of $G$, denoted by $\chi_{_{\sf FF}}(G)$, is defined as the maximum number of colors used by the First-Fit (greedy) coloring of the vertices of $G$. We call a family…

List colouring is an influential and classic topic in graph theory. We initiate the study of a natural strengthening of this problem, where instead of one list-colouring, we seek many in parallel. Our explorations have uncovered a…

Let $G$ be a planar graph without 4-cycles and 5-cycles and with maximum degree $\Delta\ge 32$. We prove that $\chi_{\ell}(G^2)\le \Delta+3$. For arbitrarily large maximum degree $\Delta$, there exist planar graphs $G_{\Delta}$ of girth 6…