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

TL;DR

This paper proves that the list chromatic number of the square of certain subcubic planar graphs without small cycles is at most 6, advancing understanding of graph coloring in restricted planar graph classes.

Contribution

It establishes that subcubic planar graphs without cycles of length 4 to 8 have a list chromatic number of their square at most 6, improving previous results.

Findings

List chromatic number of G^2 is at most 6 for these graphs.

Extends prior bounds to graphs without small cycles.

Supports conjectures on coloring properties of planar graphs.

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).