{
  "id": "1806.04628",
  "version": "v1",
  "published": "2018-06-12T16:21:08.000Z",
  "updated": "2018-06-12T16:21:08.000Z",
  "title": "The Game of Zombies and Survivors on the Cartesian Products of Trees",
  "authors": [
    "Shannon L. Fitzpatrick"
  ],
  "comment": "9 pages",
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "abstract": "We consider the game of Zombies and Survivors as introduced by Fitzpatrick, Howell, Messinger and Pike (2016) This is a variation of the game Cops and Robber where the zombies (in the cops' role) are of limited intelligence and will always choose to move closer to a survivor (who takes on the robber's role). The zombie number of a graph is defined to be the minimum number of zombies required to guarantee the capture of a survivor on the graph. In this paper, we show that the zombie number of the Cartesian product of $n$ non-trivial trees is exactly $\\lceil 2n/3 \\rceil$. This settles a conjecture by Fitzpatrick et. al. (2016) that this is the zombie number for the $n$-dimensional hypercube. In proving this result, we also discuss other variations of Cops and Robber involving active and flexible players.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-12T16:21:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "cartesian product",
      "zombie number",
      "move closer",
      "robbers role",
      "fitzpatrick"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}