{
  "id": "1004.2482",
  "version": "v1",
  "published": "2010-04-14T19:44:57.000Z",
  "updated": "2010-04-14T19:44:57.000Z",
  "title": "Variations on Cops and Robbers",
  "authors": [
    "Alan Frieze",
    "Michael Krivelevich",
    "Po-Shen Loh"
  ],
  "comment": "18 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "We consider several variants of the classical Cops and Robbers game. We treat the version where the robber can move R > 1 edges at a time, establishing a general upper bound of N / \\alpha ^{(1-o(1))\\sqrt{log_\\alpha N}}, where \\alpha = 1 + 1/R, thus generalizing the best known upper bound for the classical case R = 1 due to Lu and Peng. We also show that in this case, the cop number of an N-vertex graph can be as large as N^{1 - 1/(R-2)} for finite R, but linear in N if R is infinite. For R = 1, we study the directed graph version of the problem, and show that the cop number of any strongly connected digraph on N vertices is at most O(N(log log N)^2/log N). Our approach is based on expansion.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-04-14T19:44:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C57",
      "05C35",
      "05D40",
      "05C20"
    ],
    "keywords": [
      "cop number",
      "variations",
      "general upper bound",
      "log log",
      "robbers game"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1004.2482F"
    }
  }
}