{
  "id": "0909.3286",
  "version": "v1",
  "published": "2009-09-17T18:01:19.000Z",
  "updated": "2009-09-17T18:01:19.000Z",
  "title": "O-cycles, vertex-oriented graphs, and the four colour theore",
  "authors": [
    "Ortho Flint",
    "Stuart Rankin"
  ],
  "categories": [
    "math.CO"
  ],
  "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).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-09-17T18:01:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C10",
      "05C15"
    ],
    "keywords": [
      "planar graph",
      "vertex-oriented graphs",
      "colour theorem",
      "equivalent",
      "conceptual proof"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0909.3286F"
    }
  }
}