  The choosability $\chi_\ell(G)$ of a graph $G$ is the minimum $k$ such that
having $k$ colors available at each vertex guarantees a proper coloring. Given
a toroidal graph $G$, it is known that $\chi_\ell(G)\leq 7$, and
$\chi_\ell(G)=7$ if and only if $G$ contains $K_7$. Cai, Wang, and Zhu proved
that a toroidal graph $G$ without 7-cycles is 6-choosable, and $\chi_\ell(G)=6$
if and only if $G$ contains $K_6$. They also prove that a toroidal graph $G$
without 6-cycles is 5-choosable, and conjecture that $\chi_\ell(G)=5$ if and
only if $G$ contains $K_5$. We disprove this conjecture by constructing an
infinite family of non-4-colorable toroidal graphs with neither $K_5$ nor
cycles of length at least 6; moreover, this family of graphs is embeddable on
every surface except the plane and the projective plane. Instead, we prove the
following slightly weaker statement suggested by Zhu: toroidal graphs
containing neither $K^-_5$ (a $K_5$ missing one edge) nor 6-cycles are
4-choosable. This is sharp in the sense that forbidding only one of the two
structures does not ensure that the graph is 4-choosable.
