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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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