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List 3-dynamic coloring of graphs with small maximum average degree. (English) Zbl 1383.05106
Summary: 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 list $$r$$-dynamic chromatic number of a graph $$G$$ is denoted by $$\operatorname{ch}_r^d(G)$$.
Recently, S. Loeb et al. [Discrete Appl. Math. 235, 129–141 (2018; Zbl 1375.05098)] showed that the list 3-dynamic chromatic number of a planar graph is at most 10. J. Cheng et al. [ibid. 237, 75–81 (2018; Zbl 1380.05056)] studied the maximum average condition to have $$\chi_3^d(G) \leq 4, 5$$, or 6. On the other hand, H. Song et al. [ibid. 198, 251–263 (2016; Zbl 1327.05085)] showed that if $$G$$ is planar with girth at least 6, then $$\chi_r^d(G) \leq r + 5$$ for any $$r \geq 3$$.
In this paper, we study list 3-dynamic coloring in terms of maximum average degree. We show that $$\operatorname{ch}_3^d(G) \leq 6$$ if $$\operatorname{mad}(G) < \frac{18}{7}$$, $$\operatorname{ch}_3^d(G) \leq 7$$ if $$\operatorname{mad}(G) < \frac{14}{5}$$, and $$\operatorname{ch}_3^d(G) \leq 8$$ if $$\operatorname{mad}(G) < 3$$. All of the bounds are tight.

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C07 Vertex degrees
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##### References:
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