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

Abstract

The choosability (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 (G)7, and (G)=7 if and only if G contains K-7. Cai etal. (J Graph Theory 65(1) (2010), 1-15) proved that a toroidal graph G without 7-cycles is 6-choosable, and (G)=6 if and only if G contains K-6. They also proved that a toroidal graph G without 6-cycles is 5-choosable, and conjectured that (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 <mml:msubsup>K5-</mml:msubsup> (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. (C) 2016 Wiley Periodicals, Inc.