Squares of subcubic planar graphs without cycles of length 4-8 are 6-choosable

Abstract

The {\em square} of a graph $G$, denoted $G^2$, has the same vertex set as $G$ and an edge between any two vertices at distance at most $2$ in $G$. Wegner (1977) conjectured that for a planar graph $G$, $\chi(G^2) \leq 7$ if $\Delta(G) = 3$, $\chi(G^2) \leq \Delta(G)+5$ if $4 \leq \Delta(G) \leq 7$, and $\chi(G^2) \leq \lfloor 3\Delta(G)/2 \rfloor$ if $\Delta(G) \geq 8$, and Thomassen (2018) confirmed the conjecture for $\Delta(G) = 3$. Dvo\v{r}\'{a}k et al. (2008) and Feder et al. (2021) further conjectured that $\chi(G^2) \leq 6$ for cubic bipartite planar graphs. A natural question is whether this bound also holds for the list-chromatic number, i.e., whether $\chi_{\ell}(G^2) \leq 6$ for such graphs. More generally, it is of interest to determine sufficient conditions ensuring $\chi_{\ell}(G^2) \leq 6$ for subcubic planar graphs. In this paper, we prove that $\chi_{\ell}(G^2) \leq 6$ for subcubic planar graphs containing no $k$-cycles for $4 \leq k \leq 8$, improving a result of Cranston and Kim (2008).