Three-edge-colouring doublecross cubic graphs

TL;DR

This paper proves that every two-edge-connected doublecross cubic graph can be three-edge-coloured, confirming a special case of Tutte's conjecture using a proof method similar to the four-colour theorem.

Contribution

It solves the open doublecross case of Tutte's conjecture on three-edge-colourability of certain cubic graphs.

Findings

All two-edge-connected doublecross cubic graphs are three-edge-colourable.

The proof employs a variant of the four-colour theorem proof technique.

This result confirms a key case in Tutte's conjecture on graph colourings.

Abstract

A graph is apex if there is a vertex whose deletion makes the graph planar, and doublecross if it can be drawn in the plane with only two crossings, both incident with the infinite region in the natural sense. In 1966, Tutte conjectured that every two-edge-connected cubic graph with no Petersen graph minor is three-edge-colourable. With Neil Robertson, two of us showed that this is true in general if it is true for apex graphs and doublecross graphs. In another paper, two of us solved the apex case, but the doublecross case remained open. Here we solve the doublecross case; that is, we prove that every two-edge-connected doublecross cubic graph is three-edge-colourable. The proof method is a variant on the proof of the four-colour theorem.