{
  "id": "2204.01947",
  "version": "v1",
  "published": "2022-04-05T02:51:50.000Z",
  "updated": "2022-04-05T02:51:50.000Z",
  "title": "Tournaments and Even Graphs are Equinumerous",
  "authors": [
    "Gordon F. Royle",
    "Cheryl E. Praeger",
    "S. P. Glasby",
    "Saul D. Freedman",
    "Alice Devillers"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A graph is called \\emph{odd} if there is an orientation of its edges and an automorphism that reverses the sense of an odd number of its edges, and \\emph{even} otherwise. Pontus von Br\\\"omssen (n\\'e Andersson) showed that the existence of such an automorphism is independent of the orientation, and considered the question of counting pairwise non-isomorphic even graphs. Based on computational evidence, he made the rather surprising conjecture that the number of pairwise non-isomorphic \\emph{even graphs} on $n$ vertices is equal to the number of pairwise non-isomorphic \\emph{tournaments} on $n$ vertices. We prove this conjecture using a counting argument with several applications of the Cauchy-Frobenius Theorem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-05T02:51:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C30",
      "05C75",
      "05A15",
      "G.2.1",
      "G.2.2"
    ],
    "keywords": [
      "tournaments",
      "odd number",
      "cauchy-frobenius theorem",
      "pontus von",
      "orientation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}