Toroidal graphs containing neither $K_5^{-}$ nor 6-cycles are 4-choosable

TL;DR

This paper investigates the list coloring properties of toroidal graphs, disproves a conjecture about their choosability related to $K_5$, and proves that excluding certain subgraphs guarantees 4-choosability.

Contribution

It constructs counterexamples to a conjecture on toroidal graph choosability and proves that excluding $K_5^-$ and 6-cycles ensures 4-choosability.

Findings

Counterexamples to the $K_5$ conjecture in toroidal graphs.

Toroidal graphs without $K_5^-$ and 6-cycles are 4-choosable.

The family of graphs constructed is embeddable on all surfaces except the plane and projective plane.

Abstract

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.