{
  "id": "1708.06718",
  "version": "v1",
  "published": "2017-08-22T16:53:11.000Z",
  "updated": "2017-08-22T16:53:11.000Z",
  "title": "Simple polytopes without small separators, II: Thurston's bound",
  "authors": [
    "Lauri Loiskekoski",
    "Günter M. Ziegler"
  ],
  "comment": "10 pages, one page of figures",
  "categories": [
    "math.MG",
    "math.CO"
  ],
  "abstract": "We show that there are simple 4-dimensional polytopes with n vertices such that all separators of the graph have size at least $\\Omega(n/\\log n)$. This establishes a strong form of a claim by Thurston, for which the construction and proof had been lost. We construct the polytopes by cutting off the vertices and then the edges of a particular type of neighborly cubical polytopes. The graphs of simple polytopes thus obtained are 4-regular; they contain 3-regular \"cube-connected cycle graphs\" as minors of spanning subgraphs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-22T16:53:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52B05",
      "52B11",
      "05C12",
      "05C40"
    ],
    "keywords": [
      "simple polytopes",
      "small separators",
      "thurstons bound",
      "strong form",
      "cube-connected cycle graphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}