  A proper vertex $k$-coloring of a graph $G=(V,E)$ is an assignment $c:V\to
\{1,2,\ldots,k\}$ of colors to the vertices of the graph such that no two
adjacent vertices are associated with the same color. The square $G^2$ of a
graph $G$ is the graph defined by $V(G)=V(G^2)$ and $uv \in E(G^2)$ if and only
if the distance between $u$ and $v$ is at most two. We denote by $\chi(G^2)$
the chromatic number of $G^2$, which is the least integer $k$ such that a
$k$-coloring of $G^2$ exists. By definition, at least $\Delta(G)+1$ colors are
needed for this goal, where $\Delta(G)$ denotes the maximum degree of the graph
$G$. In this paper, we prove that the square of every graph $G$ with
$\text{mad}(G)<4$ and $\Delta(G) \geqslant 8$ is $(3\Delta(G)+1)$-choosable and
even correspondence-colorable. Furthermore, we show a family of $2$-degenerate
graphs $G$ with $\text{mad}(G)<4$, arbitrarily large maximum degree, and
$\chi(G^2)\geqslant \frac{5\Delta(G)}{2}$, improving the result of Kim and
Park.
