{
  "id": "1902.00572",
  "version": "v1",
  "published": "2019-02-01T22:08:16.000Z",
  "updated": "2019-02-01T22:08:16.000Z",
  "title": "Cycles of length three and four in tournaments",
  "authors": [
    "Timothy F. N. Chan",
    "Andrzej Grzesik",
    "Daniel Kral",
    "Jonathan A. Noel"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Linial and Morgenstern conjectured that, among all $n$-vertex tournaments with $d\\binom{n}{3}$ cycles of length three, the number of cycles of length four is asymptotically minimized by a random blow-up of a transitive tournament with all but one part of equal size and one smaller part. We prove the conjecture for $d\\ge 1/36$ by analyzing the possible spectrum of adjacency matrices of tournaments. We also demonstrate that the family of extremal examples is broader than expected and give its full description for $d\\ge 1/16$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-02-01T22:08:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "extremal examples",
      "vertex tournaments",
      "adjacency matrices",
      "smaller part",
      "random blow-up"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}