{
  "id": "1610.03924",
  "version": "v1",
  "published": "2016-10-13T03:23:31.000Z",
  "updated": "2016-10-13T03:23:31.000Z",
  "title": "Short fans and the 5/6 bound for line graphs",
  "authors": [
    "Daniel W. Cranston",
    "Landon Rabern"
  ],
  "comment": "28 pages, 5 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "In 2011, the second author conjectured that every line graph $G$ satisfies $\\chi(G)\\le \\max\\{\\omega(G),\\frac{5\\Delta(G)+8}{6}\\}$. This conjecture is best possible, as shown by replacing each edge in a 5-cycle by $k$ parallel edges, and taking the line graph. In this paper we prove the conjecture. We also develop more general techniques and results that will likely be of independent interest, due to their use in attacking the Goldberg--Seymour conjecture.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-13T03:23:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15"
    ],
    "keywords": [
      "line graph",
      "short fans",
      "goldberg-seymour conjecture",
      "independent interest",
      "parallel edges"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}