Coloring the squares of graphs whose maximum average degrees are less than 4

The square $G^2$ of a graph $G$ is the graph defined on $V(G)$ such that two vertices $u$ and $v$ are adjacent in $G^2$ if the distance between $u$ and $v$ in $G$ is at most 2. The {\em maximum average degree} of $G$, $mad (G)$, is the maximum among the average degrees of the subgraphs of $G$. It is known in \cite{BLP-14-JGT} that there is no constant $C$ such that every graph $G$ with $mad(G)< 4$ has $χ(G^2) \leq Δ(G) + C$. Charpentier \cite{Charpentier14} conjectured that there exists an integer $D$ such that every graph $G$ with $Δ(G)\ge D$ and $mad(G)<4$ has $χ(G^2) \leq 2 Δ(G)$. Recent result in \cite{BLP-DM} implies that $χ(G^2) \leq 2 Δ(G)$ if $mad(G) < 4 -\frac{1}{c}$ with $Δ(G) \geq 40c -16$. In this paper, we show for $c\ge 2$, if $mad(G) < 4 - \frac{1}{c}$ and $Δ(G) \geq 14c-7$, then $χ_\ell(G^2) \leq 2 Δ(G)$, which improves the result in \cite{BLP-DM}. We also show that for every integer $D$, there is a graph $G$ with $Δ(G)\ge D$ such that $mad(G)<4$, and $χ(G^2) \geq 2Δ(G) +2$, which disproves Charpentier's conjecture. In addition, we give counterexamples to Charpentier's another conjecture in \cite{Charpentier14}, stating that for every integer $k\ge 3$, there is an integer $D_k$ such that every graph $G$ with $mad(G)<2k$ and $Δ(G)\ge D_k$ has $χ(G^2) \leq kΔ(G) -k$.

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