Sufficient conditions on cycles that make planar graphs 4-choosable

TL;DR

This paper extends conditions under which planar graphs are 4-choosable by identifying cycle adjacency restrictions involving cycles of lengths 3 to 6.

Contribution

It generalizes previous results by establishing broader cycle adjacency conditions that guarantee 4-choosability in planar graphs.

Findings

Generalized cycle adjacency conditions for 4-choosability

Improved upon Xu and Wu's result for 5-cycles

Established new sufficient conditions involving cycles of lengths 3 to 6

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.