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,degG(v)} distinct colors in NG(v). The r-dynamic chromatic numberχrd(G) of a graph G is the least k such that there exists an r-dynamic k-coloring of G. The listr-dynamic chromatic number of a graph G is denoted by chrd(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 χ3d(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 χrd(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 ch3d(G)≤6 if mad(G)<187, ch3d(G)≤7 if mad(G)<145, and ch3d(G)≤8 if mad(G)<3. All of the bounds are tight.