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

Abstract

<abstract><p>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)\leq 10 $. This improves a result by Zhang and Li <sup>[<xref ref-type="bibr" rid="b16">16</xref>]</sup>, which says that every 1-planar graph $ G $ has $ ch_{d}(G)\leq 11 $.</p></abstract>