{
  "id": "1007.5344",
  "version": "v1",
  "published": "2010-07-29T22:30:18.000Z",
  "updated": "2010-07-29T22:30:18.000Z",
  "title": "The Radio Number of $C_n \\square C_n$",
  "authors": [
    "Marc Morris-Rivera",
    "Maggy Tomova",
    "Cindy Wyels",
    "Aaron Yeager"
  ],
  "comment": "To appear in Ars Combinatoria, 15 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Radio labeling is a variation of Hale's channel assignment problem, in which one seeks to assign positive integers to the vertices of a graph $G$ subject to certain constraints involving the distances between the vertices. Specifically, a radio labeling of a connected graph $G$ is a function $c:V(G) \\rightarrow \\mathbb Z_+$ such that $$d(u,v)+|c(u)-c(v)|\\geq 1+\\text{diam}(G)$$ for every two distinct vertices $u$ and $v$ of $G$ (where $d(u,v)$ is the distance between $u$ and $v$). The span of a radio labeling is the maximum integer assigned to a vertex. The radio number of a graph $G$ is the minimum span, taken over all radio labelings of $G$. This paper establishes the radio number of the Cartesian product of a cycle graph with itself (i.e., of $C_n\\square C_n$.)",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-07-29T22:30:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C78",
      "05C15",
      "05C38"
    ],
    "keywords": [
      "radio number",
      "radio labeling",
      "hales channel assignment problem",
      "minimum span",
      "cartesian product"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1007.5344M"
    }
  }
}