  We consider the problem of coloring the squares of graphs of bounded maximum
average degree, that is, the problem of coloring the vertices while ensuring
that two vertices that are adjacent or have a common neighbour receive
different colors.
  Borodin et al. proved in 2004 and 2008 that the squares of planar graphs of
girth at least seven and sufficiently large maximum degree $\Delta$ are list
$(\Delta+1)$-colorable, while the squares of some planar graphs of girth six
and arbitrarily large maximum degree are not. By Euler's Formula, planar graphs
of girth at least $6$ are of maximum average degree less than $3$, and planar
graphs of girth at least $7$ are of maximum average degree less than $14/5<3$.
  We strengthen their result and prove that there exists a function $f$ such
that the square of any graph with maximum average degree $m<3$ and maximum
degree $\Delta\geq f(m)$ is list $(\Delta+1)$-colorable. This bound of $3$ is
optimal in the sense that the above-mentioned planar graphs with girth $6$ have
maximum average degree less than $3$ and arbitrarily large maximum degree,
while their square cannot be $(\Delta+1)$-colored. The same holds for list
injective $\Delta$-coloring.
