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

An $r$-dynamic $k$-coloring of a graph $G$ is a proper $k$-coloring such that for any vertex $v$, there are at least $min\{r,{deg}_G\left(v\right)\}$ distinct colors in $N_G\left(v\right)$. The $r$-dynamic chromatic number${\chi }_r^d\left(G\right)$ of a graph $G$ is the least $k$ such that there exists an $r$-dynamic $k$-coloring of $G$. The list$r$-dynamic chromatic number of a graph $G$ is denoted by $c{h}_r^d\left(G\right)$.Recently, Loeb et al. (0000) showed that the list $3$-dynamic chromatic number of a planar graph is at most 10. And Cheng et al. (0000) studied the maximum average condition to have ${\chi }_3^d\left(G\right)\le 4,5$, or $6$. On the other hand, Song et al. (2016) showed that if $G$ is planar with girth at least 6, then ${\chi }_r^d\left(G\right)\le r+5$ for any $r\ge 3$.In this paper, we study list 3-dynamic coloring in terms of maximum average degree. We show that $c{h}_3^d\left(G\right)\le 6$ if $mad\left(G\right)<\frac{18}{7}$, $c{h}_3^d\left(G\right)\le 7$ if $mad\left(G\right)<\frac{14}{5}$, and $c{h}_3^d\left(G\right)\le 8$ if $mad\left(G\right)<3$. All of the bounds are tight.