  We show that for loopless $6$-regular triangulations on the torus the gap
between the choice number and chromatic number is at most $2$. We also show
that the largest gap for graphs embeddable in an orientable surface of genus
$g$ is of the order $\Theta(\sqrt{g})$, and moreover for graphs with chromatic
number of the order $o(\sqrt{g}/\log(g))$ the largest gap is of the order
$o(\sqrt{g})$.
