Sufficient conditions on cycles that make planar graphs 4-choosable

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 ${i, j, k, l} = {3,4,5,6}.$Foranychosen$i,$ifevery$i$-cycleofaplanargraph$G$isnotsimultaneouslyadjacentto$j$-cycles,$k$-cycles,and$l$-cycles, then $G$ is $4$-choosable.