Coloring the square of a sparse graph $G$ with almost $\Delta(G)$ colors

TL;DR

This paper proves new bounds on the list chromatic number of the square of a graph under certain sparsity and maximum degree conditions, advancing understanding of graph coloring in sparse and planar graphs.

Contribution

It establishes improved bounds on the list chromatic number of the square of sparse graphs, extending previous conjectures and results for large maximum degree graphs.

Findings

For large $ ext{Δ}(G)$, $ ext{ch}_ ext{ell}(G^2) o ext{Δ}(G) + c$ under certain mad conditions.

Proves that planar graphs with girth at least 5 have $ ext{χ}(G^2) o ext{Δ}(G) + 6$ for large $ ext{Δ}(G)$.

Abstract

For a graph $G$, let $G^2$ be the graph with the same vertex set as $G$ and $xy \in E(G^2)$ when $x \neq y$ and $d_G(x,y) \leq 2$. Bonamy, L\'ev\^{e}que, and Pinlou conjectured that if $mad (G) < 4 - \frac{2}{c+1}$ and $\Delta(G)$ is large, then $\chi_\ell(G^2) \leq \Delta(G) + c$. We prove that if $c \geq 3$, $mad (G) < 4 - \frac{4}{c+1}$, and $\Delta(G)$ is large, then $\chi_\ell(G^2) \leq \Delta(G) + c$. Dvo\v{r}\'ak, Kr\'{a}\soft{l}, Nejedl\'{y}, and \v{S}krekovski conjectured that $\chi(G^2) \leq \Delta(G) +2$ when $\Delta(G)$ is large and $G$ is planar with girth at least $5$; our result implies $\chi (G^2) \leq \Delta(G) +6$.