O-cycles, vertex-oriented graphs, and the four colour theore
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).
Cite
@article{arxiv.0909.3286,
title = {O-cycles, vertex-oriented graphs, and the four colour theore},
author = {Ortho Flint and Stuart Rankin},
journal= {arXiv preprint arXiv:0909.3286},
year = {2009}
}