Czechoslovak Mathematical Journal, Vol. 70, No. 1, pp. 161-178, 2020
Acyclic 4-choosability of planar graphs without 4-cycles
Yingcai Sun, Min Chen
Received April 9, 2019. Published online September 16, 2019.
Abstract: A proper vertex coloring of a graph $G$ is acyclic if there is no bicolored cycle in $G$. In other words, each cycle of $G$ must be colored with at least three colors. Given a list assignment $L=\{L(v)\colon v\in V\}$, if there exists an acyclic coloring $\pi$ of $G$ such that $\pi(v)\in L(v)$ for all $v\in V$, then we say that $G$ is acyclically $L$-colorable. If $G$ is acyclically $L$-colorable for any list assignment $L$ with $|L(v)|\ge k$ for all $v\in V$, then $G$ is acyclically $k$-choosable. In 2006, Montassier, Raspaud and Wang conjectured that every planar graph without 4-cycles is acyclically 4-choosable. However, this has been as yet verified only for some restricted classes of planar graphs. In this paper, we prove that every planar graph with neither 4-cycles nor intersecting $i$-cycles for each $i\in\{3,5\}$ is acyclically 4-choosable.
Affiliations: Yingcai Sun, Min Chen (corresponding author), Department of Mathematics, Zhejiang Normal University, 688 Yingbin Avenue, Jinhua 321004, P. R. China, e-mail: chenmin@zjnu.cn