Dynamic coloring and list dynamic coloring of planar graphs

A dynamic chromatic number χd(G) of a graph G is the least number k such that G has a proper k-coloring of the vertex set V (G) so that for each vertex of degree at least 2, its neighbors receive at least two distinct colors. We show that χd(G) ≤ 4 for every planar graph except C5, which was conjectured in [5]. The list dynamic chromatic number chd(G) of G is the least number k such that for any assignment of k-element lists to the vertices of G, there is a dynamic coloring of G where the color on each vertex is chosen from its list. Based on Thomassen’s result [12] 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) ≤ 5 for every planar graph.