  Motivated by the question of whether planar graphs have bounded queue-number,
we prove that planar graphs with maximum degree $\Delta$ have queue-number
$O(\Delta^{2})$, which improves upon the best previous bound of $O(\Delta^6)$.
More generally, we prove that graphs with bounded degree and bounded Euler
genus have bounded queue-number. In particular graphs with Euler genus $g$ and
maximum degree $\Delta$ have queue-number $O(g+\Delta^{2})$. As a byproduct we
prove that if planar graphs have bounded queue-number, then graphs of Euler
genus $g$ have queue-number $O(g)$.
