# Planar graphs without pairwise adjacent 3-,4-,5-, and 6-cycle are 4-choosable

## TL;DR

This paper proves that planar graphs lacking pairwise adjacent 3-, 4-, 5-, and 6-cycles are 4-choosable, extending previous results by Xu and Wu to a broader class of graphs.

## Contribution

It establishes a new sufficient condition for 4-choosability in planar graphs, generalizing earlier work by removing adjacency restrictions among certain cycles.

## Key findings

- Planar graphs without pairwise adjacent 3-, 4-, 5-, and 6-cycles are 4-choosable.
- The result broadens the class of graphs known to be 4-choosable.
- Improves upon previous conditions requiring non-adjacency of 5-cycles to 3- and 4-cycles.

## 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. If G is a planar graph without pairwise adjacent 3-,4-,5-, and 6-cycle, then G is 4-choosable.