  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 $\chi(G^2) \leq \Delta(G) + C$.
Charpentier \cite{Charpentier14} conjectured that there exists an integer $D$
such that every graph $G$ with $\Delta(G)\ge D$ and $mad(G)<4$ has $\chi(G^2)
\leq 2 \Delta(G)$. Recent result in \cite{BLP-DM} implies that $\chi(G^2) \leq
2 \Delta(G)$ if $mad(G) < 4 -\frac{1}{c}$ with $\Delta(G) \geq 40c -16$.
  In this paper, we show for $c\ge 2$, if $mad(G) < 4 - \frac{1}{c}$ and
$\Delta(G) \geq 14c-7$, then $\chi_\ell(G^2) \leq 2 \Delta(G)$, which improves
the result in \cite{BLP-DM}. We also show that for every integer $D$, there is
a graph $G$ with $\Delta(G)\ge D$ such that $mad(G)<4$, and $\chi(G^2) \geq
2\Delta(G) +2$, 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 $\Delta(G)\ge D_k$ has $\chi(G^2) \leq
k\Delta(G) -k$.
