1-Planar graphs without 6-cycles are 6-choosable

Abstract

A graph is 1-planar if it can be drawn in the plane so that each edge is crossed by at most one other edge. A graph is $k$-degenerate if each of its subgraphs contains a vertex of degree at most $k$. It was known that every 1-planar graph is 8-choosable. In this paper, we show that every 1-planar graph without 6-cycles is 5-degenerate and hence 6-choosable.