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

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, Lévêque, and Pinlou conjectured that if $mad (G) < 4 - \frac{2}{c+1}$ and $Δ(G)$ is large, then $χ_\ell(G^2) \leq Δ(G) + c$. We prove that if $c \geq 3$, $mad (G) < 4 - \frac{4}{c+1}$, and $Δ(G)$ is large, then $χ_\ell(G^2) \leq Δ(G) + c$. Dvořák, Krá\soft{l}, Nejedlý, and Škrekovski conjectured that $χ(G^2) \leq Δ(G) +2$ when $Δ(G)$ is large and $G$ is planar with girth at least $5$; our result implies $χ(G^2) \leq Δ(G) +6$.

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