# Some conditions on 5-cycles that make planar graphs 4-choosable

## TL;DR

This paper identifies specific structural conditions on planar graphs involving 5-cycles that guarantee the graph is 4-choosable, expanding understanding of list coloring in planar graphs.

## Contribution

It introduces new conditions on 5-cycles that ensure planar graphs are 4-choosable, broadening previous results in graph coloring theory.

## Key findings

- Planar graphs with certain 5-cycle conditions are 4-choosable
- Graphs where 5-cycles are not adjacent to 3-cycles are 4-choosable
- Structural restrictions on 5-cycles influence list coloring properties

## Abstract

Consider two conditions on a graph: (1) each 5-cycle is not a subgraph of 5-wheel and does not share exactly one edge with 3-cycle, and (2) each 5-cycle is not adjacent to two 3-cycles and is not adjacent to a 4-cycle with chord. We show that if a planar graph $G$ satisfies one of the these conditions, then $G$ is 4-choosable. This yields that if each 5-cycle of a planar graph $G$ is not adjacent a 3-cycle, then $G$ is 4-choosable.