{
  "id": "math/0603504",
  "version": "v1",
  "published": "2006-03-21T13:38:39.000Z",
  "updated": "2006-03-21T13:38:39.000Z",
  "title": "On the size of spheres of relations with a transitive group of automorphisms",
  "authors": [
    "Yahya Ould Hamidoune"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $\\Gamma =(V,E)$ be a point-transitive reflexive relation. Let $v\\in V$ and put $r=|\\Gamma (v)|.$ Also assume $\\Gamma ^j(v)\\cap \\Gamma ^{-}(v)=\\{v\\}$. Then $$ |\\Gamma ^{j} (v)\\setminus \\Gamma ^{j-1} (v)| \\ge r-1.$$ In particular we have $ |\\Gamma ^{j} (v)| \\ge 1+(r-1)j.$ The last result confirms a recent conjecture of Seymour in the case vertex-transitive graphs. Also it gives a short proof for the validity of the Caccetta-H\\\"aggkvist conjecture for vertex-transitive graphs and generalizes an additive result of Shepherdson.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-03-21T13:38:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20D60"
    ],
    "keywords": [
      "transitive group",
      "automorphisms",
      "conjecture",
      "result confirms",
      "case vertex-transitive graphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......3504O"
    }
  }
}