The square of every subcubic planar graph without 4-cycles and 5-cycles is 7-choosable

TL;DR

This paper proves that the list chromatic number of the square of certain subcubic planar graphs without 4- and 5-cycles is at most 7, extending previous results on coloring such graphs.

Contribution

It establishes that subcubic planar graphs without 4- and 5-cycles have their square's list chromatic number bounded by 7, improving prior bounds for graphs with girth at least 6.

Findings

List chromatic number of G^2 is at most 7 for specified graphs.

Extends previous results from girth ≥ 6 to graphs without 4- and 5-cycles.

Improves understanding of coloring properties of subcubic planar graphs.

Abstract

The square of a graph $G$, denoted by $G^2$, has the same vertex set as $G$ and has an edge between two vertices if the distance between them in $G$ is at most $2$. Thomassen (2018) and independently, Hartke, Jahanbekam and Thomas (2016) proved that $\chi(G^2) \leq 7$ if $G$ is a subcubic planar graph. A natural question is whether $\chi_{\ell}(G^2) \leq 7$ or not if $G$ is a subcubic planar graph. Recently, Kim and Lian (2024) proved that $\chi_{\ell}(G^2) \leq 7$ if $G$ is a subcubic planar graph of girth at least 6. In this paper, we prove that $\chi_{\ell}(G^2) \leq 7$ if $G$ is a subcubic planar graph without 4-cycles and 5-cycles, which improves the result of Kim and Lian.