$k$-forested coloring of planar graphs with large girth

Abstract

A proper vertex coloring of a simple graph $G$ is $k$-forested if the subgraph induced by the vertices of any two color classes is a forest with maximum degree at most $k$. The $k$-forested chromatic number of a graph $G$, denoted by $\chi^{a}_{k}(G)$, is the smallest number of colors in a $k$-forested coloring of $G$. In this paper, it is shown that planar graphs with large enough girth do satisfy $\chi^{a}_{k}(G)=\lceil\frac{\Delta(G)}{k}\rceil+1$ for all $\Delta(G)> k\geq 2$, and $\chi^{a}_{k}(G)\leq 3$ for all $\Delta(G)\leq k$ with the bound 3 being sharp. Furthermore, a conjecture on $k$-frugal chromatic number raised in [1] has been partially confirmed.