Sufficient conditions on cycles that make planar graphs 4-choosable

Abstract

Xu and Wu proved that if every $5$-cycle of a planar graph $G$ is not simultaneously adjacent to $3$-cycles and $4$-cycles, then $G$ is $4$-choosable. In this paper, we improve this result as follows. Let $\{i,j,k,l\}=\{3,4,5,6\}.$ For any chosen $i,$ if every $i$-cycle of a planar graph $G$ is not simultaneously adjacent to $j$-cycles, $k$-cycles, and $l$-cycles, then $G$ is $4$-choosable.