Revisiting a Nice Cycle Lemma and its Consequences

TL;DR

This paper corrects errors in previous works related to graph colorings on surfaces and provides a concise proof that certain triangulations are 5-colorable, leveraging the Four Color Theorem for simplicity.

Contribution

It offers corrections to earlier proofs and introduces a shorter proof of Thomassen's theorem on 5-colorability of surface triangulations with long noncontractible cycles.

Findings

Corrected errors in prior graph theory papers.

Provided a shorter proof of Thomassen's 5-colorability theorem.

Utilized the Four Color Theorem in the proof, simplifying the original argument.

Abstract

We correct some errors and omissions primarily in a paper [Albertson&Hutchinson2004], discovered by R.B. Richter, and also some in a proof of [Thomassen1993] and of [Yu1997]. We give a short proof of Thomassen's theorem that every triangulation of a surface with all noncontractible cycles sufficiently long can be 5-colored; part of the shortness is due to the use of the Four Color Theorem, which is not used in Thomassen's original proof.