The square of every subcubic planar graph of girth at least 6 is 7-choosable

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 (2018) and 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. Cranston and Kim (2008) showed that ${\chi }_{\ell }(G^2)\le 7$ if $G$ is a subcubic planar graph of girth at least 7. We prove that ${\chi }_{\ell }(G^2)\le 7$ if $G$ is a subcubic planar graph of girth at least 6.

9 pages, 1 figure