  Let $G$ be a planar graph without 4-cycles and 5-cycles and with maximum
degree $\Delta\ge 32$. We prove that $\chi_{\ell}(G^2)\le \Delta+3$. For
arbitrarily large maximum degree $\Delta$, there exist planar graphs
$G_{\Delta}$ of girth 6 with $\chi(G_{\Delta}^2)=\Delta+2$. Thus, our bound is
within 1 of being optimal. Further, our bound comes from coloring greedily in a
good order, so the bound immediately extends to online list-coloring. In
addition, we prove bounds for $L(p,q)$-labeling. Specifically,
$\lambda_{2,1}(G)\le \Delta+8$ and, more generally, $\lambda_{p,q}(G)\le
(2q-1)\Delta+6p-2q-2$, for positive integers $p$ and $q$ with $p\ge q$. Again,
these bounds come from a greedy coloring, so they immediately extend to the
list-coloring and online list-coloring variants of this problem.
