Related papers: On sufficient conditions for planar graphs to be 5…

List coloring generalizes graph coloring by requiring the color of a vertex to be selected from a list of colors specific to that vertex. One refinement of list coloring, called choosability with separation, requires that the intersection…

A vertex coloring of a graph G is called a 2-distance coloring if any two vertices at a distance at most 2 from each other receive different colors. Suppose that G is a planar graph with a maximum degree at most 5. We prove that G admits a…

Let $G$ be an $n$-vertex graph and let $L:V(G)\rightarrow P(\{1,2,3\})$ be a list assignment over the vertices of $G$, where each vertex with list of size 3 and of degree at most 5 has at least three neighbors with lists of size 2. We can…

A \emph{request} on a graph assigns a preferred color to a subset of the vertices. A graph $G$ is \emph{$\epsilon$-flexibly $k$-choosable} if for every $k$-list assignment $L$ and every request $r$ on $G$, there is an $L$-coloring such that…

Given two graphs $H_1$ and $H_2$, a graph $G$ is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ or $H_2$. A $P_t$ is the path on $t$ vertices. A chair is a $P_4$ with an additional vertex adjacent to one of the…

A list assignment $L$ of a graph $G$ is a function that assigns to every vertex $v$ of $G$ a set $L(v)$ of colors. A proper coloring $\alpha$ of $G$ is called an $L$-coloring of $G$ if $\alpha(v)\in L(v)$ for every $v\in V(G)$. For a list…

We study squares of planar graphs with the aim to determine their list chromatic number. We present new upper bounds for the square of a planar graph with maximum degree $\Delta \leq 4$. In particular $G^2$ is 5-, 6-, 7-, 8-, 12-,…

For a number $\ell\geq 2$, let $\mathcal{H}_{\ell}$ denote the family of graphs which have girth $2\ell$ and have no even hole with length greater than $2\ell$. Wu, Xu, and Xu conjectured that every graph in…

A {\em conflict-free coloring} of a graph {\em with respect to open} (resp., {\em closed}) {\em neighborhood} is a coloring of vertices such that for every vertex there is a color appearing exactly once in its open (resp., closed)…

We consider (not necessarily proper) colorings of the vertices of a graph where every color is thoroughly distributed, that is, appears in every open neighborhood. Equivalently, every color is a total dominating set. We define $\td(G)$ as…

A {\it list assignment} $L$ of a graph $G$ is a function that assigns a set (list) $L(v)$ of colors to every vertex $v$ of $G$. Graph $G$ is called {\it $L$-list colorable} if it admits a vertex coloring $\phi$ such that $\phi(v)\in L(v)$…

Let $\partial_H(u)$ be the set of edges incident with a vertex $u$ in the graph $H$. We say that a graph $G$ is $H$-colorable if there exist total functions $f : E(G) \rightarrow E(H)$ and $g : V(G) \rightarrow V(H)$ such that $f$ is a…

For planar graphs, we consider the problems of \emph{list edge coloring} and \emph{list total coloring}. Edge coloring is the problem of coloring the edges while ensuring that two edges that are adjacent receive different colors. Total…

The 3-coloring of hereditary graph classes has been a deeply-researched problem in the last decade. A hereditary graph class is characterized by a (possibly infinite) list of minimal forbidden induced subgraphs $H_1,H_2,\ldots$; the graphs…

For a set of nonnegative integers $c_1, \ldots, c_k$, a $(c_1, c_2,\ldots, c_k)$-coloring of a graph $G$ is a partition of $V(G)$ into $V_1, \ldots, V_k$ such that for every $i$, $1\le i\le k, G[V_i]$ has maximum degree at most $c_i$. We…

In 1965, Vizing proved that every planar graph $G$ with maximum degree $\Delta\geq 8$ is edge $\Delta$-colorable. It is also proved that every planar graph $G$ with maximum degree $\Delta=7$ is edge $\Delta$-colorable by Sanders and Zhao,…

This paper proves that for each positive integer $m$, there is a planar graph $G$ which is not $(4m+\lfloor \frac{2m-1}{9}\rfloor,m)$-choosable. Then we pose some conjectures concerning multiple list colouring of planar graphs.

Thomassen proved that all planar graphs are $5$-choosable. \v{S}krekovski strengthened the result by showing that all $K_{5}$-minor-free graphs are $5$-choosable. Dvo\v{r}\'{a}k and Postle pointed out that all planar graphs are…

Flexible list coloring was introduced by Dvo\v{r}\'{a}k, Norin, and Postle in 2019. Suppose $0 \leq \epsilon \leq 1$, $G$ is a graph, $L$ is a list assignment for $G$, and $r$ is a function with non-empty domain $D\subseteq V(G)$ such that…

We prove that the two-colouring number of any planar graph is at most 8. This resolves a question of Kierstead et al. [SIAM J. Discrete Math.~23 (2009), 1548--1560]. The result is optimal.