The $r$-acyclic chromatic number of planar graphs

Abstract

A vertex coloring of a graph G is r-acyclic if it is a proper vertex coloring such that every cycle $C$ receives at least $\min \{|C|,r\}$ colors. The $r$-acyclic chromatic number $a_{r}(G)$ of $G$ is the least number of colors in an $r$-acyclic coloring of $G$. Let $G$ be a planar graph. By Four Color Theorem, we know that $a_{1}(G)=a_{2}(G)=\chi (G)\le 4$, where $\chi (G)$ is the chromatic number of $G$. Borodin proved that $a_{3}(G)\le 5$. However when $r\ge 4$, the $r$-acyclic chromatic number of a class of graphs may not be bounded by a constant number. For example, $a_{4}(K_{2,n})=n+2=\Delta (K_{2,n})+2$ for $n\ge 2$, where $K_{2,n}$ is a complete bipartite (planar) graph. In this paper, we give a sufficient condition for $a_{r}(G)\le r$ when $G$ is a planar graph. In precise, we show that if $r\ge 4$ and $G$ is a planar graph with $g(G)\ge \frac{10r-4}{3}$, then $a_{r}(G)\le r$. In addition, we discuss the $4$-acyclic colorings of some special planar graphs.