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 $c{h}_d(G)$ 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 $c{h}_d(G)\le 10$. This improves a result by Zhang and Li ^[16], which says that every 1-planar graph $G$ has $c{h}_d(G)\le 11$.