O-cycles, vertex-oriented graphs, and the four colour theore

TL;DR

This paper introduces vertex-oriented 4-regular planar graphs derived from 3-regular graphs and shows their o-colourability is equivalent to the four colour theorem, offering a new approach to its proof.

Contribution

It defines vertex-oriented 4-regular planar graphs and establishes their o-colourability equivalence to the four colour theorem, proposing a novel proof strategy.

Findings

Every VOGWOC is o-colourable.

The four colour theorem is equivalent to the 3-o-colourability of VOGWOC.

An alternative approach to prove the four colour theorem is proposed.

Abstract

In 1880, P. G. Tait showed that the four colour theorem is equivalent to the assertion that every 3-regular planar graph without cut-edges is 3-edge-colourable, and in 1891, J. Petersen proved that every 3-regular graph with at most two cut-edges has a 1-factor. In this paper, we introduce the notion of collapsing all edges of a 1-factor of a 3-regular planar graph, thereby obtaining what we call a vertex-oriented 4-regular planar graph. We also introduce the notion of o-colouring a vertex-oriented 4-regular planar graph, and we prove that the four colour theorem is equivalent to the assertion that every vertex-oriented 4-regular planar graph without nontransversally oriented cut-vertex (VOGWOC in short) is 3-o-colourable. This work proposes an alternative avenue of investigation in the search to find a more conceptual proof of the four colour theorem, and we are able to prove that every VOGWOC is o-colourable (although we have not yet been able to prove 3-o-colourability).