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 $G$, $χ(G^2) \leq 7$ if $Δ(G) = 3$, $χ(G^2) \leq Δ(G)+5$ if $4 \leq Δ(G) \leq 7$, and $χ(G^2) \leq \lfloor 3Δ(G)/2 \rfloor$ if $Δ(G) \geq 8$, and Thomassen (2018) confirmed the conjecture for $Δ(G) = 3$. Dvořák et al. (2008) and Feder et al. (2021) further conjectured that $χ(G^2) \leq 6$ for cubic bipartite planar graphs. A natural question is whether this bound also holds for the list-chromatic number, i.e., whether $χ_{\ell}(G^2) \leq 6$ for such graphs. More generally, it is of interest to determine sufficient conditions ensuring $χ_{\ell}(G^2) \leq 6$ for subcubic planar graphs. In this paper, we prove that $χ_{\ell}(G^2) \leq 6$ for subcubic planar graphs containing no $k$-cycles for $4 \leq k \leq 8$, improving a result of Cranston and Kim (2008).