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,degG(v)} distinct colors in NG(v). The r-dynamic chromatic number χr 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 chr d(G). 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 χ3 d(G)≤4,5, or 6. On the other hand, Song et al. (2016) showed that if G is planar with girth at least 6, then χr d(G)≤r+5 for any r≥3. In this paper, we study list 3-dynamic coloring in terms of maximum average degree. We show that ch3 d(G)≤6 if mad(G)<[Formula presented], ch3 d(G)≤7 if mad(G)<[Formula presented], and ch3 d(G)≤8 if mad(G)<3. All of the bounds are tight.
We would like to thank the two anonymous reviewers for helpful and valuable comments. The first author\u2019s research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education ( NRF-2015R1D1A1A01057008 ). The second author\u2019s research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning ( 2015R1C1A1A01053495 ).
