The square of a subcubic planar graph without a 5-cycle is 7-choosable

TL;DR

This paper proves that the square of a subcubic planar graph without 5-cycles is 7-choosable, extending previous results on coloring such graphs and their list colorings.

Contribution

It establishes that the square of a subcubic planar graph without 5-cycles is 7-choosable, improving upon prior bounds for graphs with certain girth and cycle restrictions.

Findings

The square of such graphs is 7-choosable.

This result generalizes previous bounds for graphs with girth at least 6.

It advances understanding of list coloring in planar graphs.

Abstract

The square of a graph $G$, denoted $G^2$, 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 [12] showed 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 [11] showed that $\chi_{\ell}(G^2) \leq 7$ if $G$ is a subcubic planar graph of girth at least 6. And Jin, Kang, and Kim [10] showed that $\chi_{\ell}(G^2) \leq 7$ if $G$ is a subcubic planar graph without 4-cycles and 5-cycles. In this paper, we show that the square of a subcubic planar graph without 5-cycles is 7-choosable, which improves the results of [10] and [11].