Acyclic 4-choosability of planar graphs

Abstract

A proper vertex coloring of a graph $G=(V,E)$ is acyclic if $G$ contains no bicolored cycle. Given a list assignment $L=\{L\left(v\right)\mid v\in V\}$ of $G$, we say $G$ is acyclically $L$-list colorable if there exists a proper acyclic coloring $\pi$ of $G$ such that $\pi \left(v\right)\in L\left(v\right)$ for all $v\in V$. If $G$ is acyclically $L$-list colorable for any list assignment with $\left|L\left(v\right)\right|\ge k$ for all $v\in V$, then $G$ is acyclically $k$-choosable. In this paper we prove that planar graphs without 4, 7, and 8-cycles are acyclically 4-choosable.