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We study squares of planar graphs with the aim to determine their listchromatic number. We present new upper bounds for the square of a planar graphwith 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, 3respectively. In fact we prove more general results, in terms of maximumaverage degree, that imply the results above.

Choosability of the square of a planar graph with maximum degree four