  We study squares of planar graphs with the aim to determine their list
chromatic number. We present new upper bounds for the square of a planar graph
with maximum degree $\Delta \leq 4$. In particular $G^2$ is 5-, 6-, 7-, 8-,
12-, 14-choosable if the girth of $G$ is at least 16, 11, 9, 7, 5, 3
respectively. In fact we prove more general results, in terms of maximum
average degree, that imply the results above.

Combinatorics