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

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 $\chi (G^2)\le 7$ if $G$ is a subcubic planar graph. A natural question is whether ${\chi }_{\ell }(G^2)\le 7$ or not if $G$ is a subcubic planar graph. Recently, Kim and Lian (2024) proved that ${\chi }_{\ell }(G^2)\le 7$ if $G$ is a subcubic planar graph of girth at least 6. In this paper, we prove that ${\chi }_{\ell }(G^2)\le 7$ if $G$ is a subcubic planar graph without 4-cycles and 5-cycles, which improves the result of Kim and Lian.

10 pages