Toroidal graphs containing neither nor 6‐cycles are 4‐choosable

Abstract

AbstractThe choosability 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 , and if and only if G contains K7. Cai et al. (J Graph Theory 65(1) (2010), 1–15) proved that a toroidal graph G without 7‐cycles is 6‐choosable, and if and only if G contains K6. They also proved that a toroidal graph G without 6‐cycles is 5‐choosable, and conjectured that 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 (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.