Dynamic coloring and list dynamic coloring of planar graphs

A dynamic coloring   of a graph GG is a proper coloring of the vertex set V(G)V(G) such that for each vertex of degree at least 2, its neighbors receive at least two distinct colors. The dynamic chromatic number  χd(G)χd(G) of a graph GG is the least number kk such that GG has a dynamic coloring with kk colors. We show that χd(G)≤4χd(G)≤4 for every planar graph except C5C5, which was conjectured in Chen et al. (2012) [5].The list dynamic chromatic number  chd(G)chd(G) of GG is the least number kk such that for any assignment of kk-element lists to the vertices of GG, there is a dynamic coloring of GG where the color on each vertex is chosen from its list. Based on Thomassen’s (1994) result [14] that every planar graph is 5-choosable, an interesting question is whether the list dynamic chromatic number of every planar graph is at most 5 or not. We answer this question by showing that chd(G)≤5chd(G)≤5 for every planar graph.