Toroidal graphs containing neither K−5 nor 6-cycles are 4-choosable

The choosability χ`(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 χ`(G) ≤ 7, and χ`(G) = 7 if and only if G contains K7. Cai, Wang, and Zhu proved that a toroidal graph G without 7-cycles is 6-choosable, and χ`(G) = 6 if and only if G contains K6. They also prove that a toroidal graph G without 6-cycles is 5-choosable, and conjecture that χ`(G) = 5 if and only if G contains K5. We disprove this conjecture by constructing an infinite family of non-4-colorable toroidal graphs with neither K5 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 K5 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.