{
  "id": "1812.07327",
  "version": "v1",
  "published": "2018-12-18T12:28:33.000Z",
  "updated": "2018-12-18T12:28:33.000Z",
  "title": "1-subdivisions, fractional chromatic number and Hall ratio",
  "authors": [
    "Zdeněk Dvořák",
    "Patrice Ossona de Mendez",
    "Hehui Wu"
  ],
  "comment": "12 pages, no figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "The Hall ratio of a graph G is the maximum of |V(H)|/alpha(H) over all subgraphs H of G. Clearly, the Hall ratio of a graph is a lower bound for the fractional chromatic number. It has been asked whether conversely, the fractional chromatic number is upper bounded by a function of the Hall ratio. We answer this question in negative, by showing two results of independent interest regarding 1-subdivisions (the 1-subdivision of a graph is obtained by subdividing each edge exactly once). * For every c > 0, every graph of sufficiently large average degree contains as a subgraph the 1-subdivision of a graph of fractional chromatic number at least c. * For every d > 0, there exists a graph G of average degree at least d such that every graph whose 1-subdivision appears as a subgraph of G has Hall ratio at most 18. We also discuss the consequences of these results in the context of graph classes with bounded expansion.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-18T12:28:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15",
      "G.2.2"
    ],
    "keywords": [
      "fractional chromatic number",
      "hall ratio",
      "sufficiently large average degree contains",
      "independent interest"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}