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We consider the problem of coloring the squares of graphs of bounded maximumaverage degree, that is, the problem of coloring the vertices while ensuringthat two vertices that are adjacent or have a common neighbour receivedifferent colors.  Borodin et al. proved in 2004 and 2008 that the squares of planar graphs ofgirth at least seven and sufficiently large maximum degree $\Delta$ are list$(\Delta+1)$-colorable, while the squares of some planar graphs of girth sixand arbitrarily large maximum degree are not. By Euler's Formula, planar graphsof girth at least $6$ are of maximum average degree less than $3$, and planargraphs 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 $f$ suchthat the square of any graph with maximum average degree $m<3$ and maximumdegree $\Delta\geq f(m)$ is list $(\Delta+1)$-colorable. This bound of $3$ isoptimal in the sense that the above-mentioned planar graphs with girth $6$ havemaximum average degree less than $3$ and arbitrarily large maximum degree,while their square cannot be $(\Delta+1)$-colored. The same holds for listinjective $\Delta$-coloring.

List coloring the square of sparse graphs with large degree