On list 3-dynamic coloring of near-triangulations

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). Loeb et al. (2018) showed that ch3d(G)≤10 for every planar graph G, and there is a planar graph G with χ3d(G)=7.In this paper, we study a special class of planar graphs which have better upper bounds of ch3d(G). We prove that ch3d(G)≤6 if G is a planar graph which is a near-triangulation, where a near-triangulation is a planar graph whose bounded faces are all 3-cycles.