An improved upper bound for the dynamic list coloring of 1-planar graphs

A graph is 1-planar if it can be drawn in the plane such that each of its edges is crossed at most once. A dynamic coloring of a graph G is a proper vertex coloring such that for each vertex of degree at least 2, its neighbors receive at least two different colors. The list dynamic chromatic number ch(d)(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. In this paper, we show that if G is a 1-planar graph, then ch(d)(G) <= 10. This improves a result by Zhang and Li [16], which says that every 1-planar graph G has ch(d)(G) <= 11.