Seog-Jin Kim (김석진), The square of every subcubic planar graph of girth at least 6 is 7-choosable

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 $2$. Wegner’s conjecture (1977) states that for a planar graph $G$, the chromatic number $\chi (G^2)$ of $G^2$ is at most 7 if $\mathrm{\Delta }(G)=3$, at most $\mathrm{\Delta }(G)+5$ if $4\le \mathrm{\Delta }(G)\le 7$, and at most $\lfloor \frac{3\mathrm{\Delta }(G)}{2}\rfloor$ if $\mathrm{\Delta }(G)\ge 8$. Wegner’s conjecture is still wide open. The only case for which we know tight bound is when $\mathrm{\Delta }(G)=3$. Thomassen (2018) showed that $\chi (G^2)\le 7$ if $G$ is a planar graph with $\mathrm{\Delta }(G)=3$, which implies that Wegner’s conjecture is true for $\mathrm{\Delta }(G)=3$. A natural question is whether ${\chi }_{\ell }(G^2)\le 7$ or not if $G$ is a subcubic planar graph, where ${\chi }_{\ell }(G^2)$ is the list chromatic number of $G^2$. 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. This is joint work with Xiaopan Lian (Nankai University).