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.