List 3-dynamic coloring of graphs with small maximum average degree

Abstract

An $r$-dynamic $k$-coloring of a graph $G$ is a proper $k$-coloring such that for any vertex $v$, there are at least $\min\{r,\deg_G(v) \}$ distinct colors in $N_G(v)$. The $r$-dynamic chromatic number $\chi_r^d(G)$ of a graph $G$ is the least $k$ such that there exists an $r$-dynamic $k$-coloring of $G$. The {\em list $r$-dynamic chromatic number} of a graph $G$ is denoted by $ch_r^d(G)$. Recently, Loeb et al. [UI] showed that the list $3$-dynamic chromatic number of a planar graph is at most 10. And Cheng et al. [Lai-16] studied the maximum average condition to have $\chi_3^d (G) \leq 4, \ 5$, or $6$. On the other hand, Song et al. [SLW] showed that if $G$ is planar with girth at least 6, then $\chi_r^d(G)\le r+5$ for any $r\ge 3$. In this paper, we study list 3-dynamic coloring in terms of maximum average degree. We show that $ch^d_3(G) \leq 6$ if $mad(G) < \frac{18}{7}$, $ch^d_3(G) \leq 7$ if $mad(G) < \frac{14}{5}$, and $ch^d_3(G) \leq 8$ if $mad(G) < 3$. All of the bounds are tight.