{
  "id": "1911.13243",
  "version": "v1",
  "published": "2019-11-29T17:40:37.000Z",
  "updated": "2019-11-29T17:40:37.000Z",
  "title": "Distance domatic numbers for grid graphs",
  "authors": [
    "Alex Cameron",
    "Jiasheng Yan"
  ],
  "comment": "27 pages, 12 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "We say that a vertex-coloring of a graph is a proper k-distance domatic coloring if for each color, every vertex is within distance k from a vertex receiving that color. The maximum number of colors for which such a coloring exists is called the k-distance domatic number of the graph. The problem of determining the k-distance domatic number is motivated by questions about multi-agent networks including arrangements of sensors and robotics. Here, we find the exact k-distance domatic numbers for all grid graphs formed from the Cartesian product of two sufficiently long paths.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-29T17:40:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "grid graphs",
      "exact k-distance domatic numbers",
      "proper k-distance domatic coloring",
      "maximum number",
      "cartesian product"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}