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

Paper

Abstract

The {\em square} of a graph

, denoted

, has the same vertex set as

and an edge between any two vertices at distance at most

. Wegner (1977) conjectured that for a planar graph

, and

, and Thomassen (2018) confirmed the conjecture for

. Dvořák et al. (2008) and Feder et al. (2021) further conjectured that

for cubic bipartite planar graphs. A natural question is whether this bound also holds for the list-chromatic number, i.e., whether

for such graphs. More generally, it is of interest to determine sufficient conditions ensuring

for subcubic planar graphs. In this paper, we prove that

for subcubic planar graphs containing no

-cycles for

, improving a result of Cranston and Kim (2008).