List Dynamic Coloring of Sparse Graphs

Abstract

A dynamic coloring of a graph G is a proper coloring of the vertex set V(G) such that each vertex neighborhood of size at least 2 receives at least two distinct colors. The list dynamic chromatic number ch _d (G) of G is the least integer k such that for every list assignment of size k to each vertex of G, there is a dynamic coloring of G such that each vertex is colored by a color from its list. We proved that ch _d (G) ≤ 4 if $\mathrm{Mad} (G) < \frac{8}{3}$ where Mad (G) is the maximum average degree of G. And ch _d (G) ≤ 4 if G is a planar graph of girth at least 7. Both results are sharp. In addition, we show that ch _d (G) ≤ 6 for every planar graph G.