{
  "id": "2403.19555",
  "version": "v1",
  "published": "2024-03-28T16:38:26.000Z",
  "updated": "2024-03-28T16:38:26.000Z",
  "title": "On 5-cycles and strong 5-subtournaments in a tournament of odd order n",
  "authors": [
    "Sergey Savchenko"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $T$ be a tournament of odd order $n\\ge 5,$ $c_{m}(T)$ be the number of its $m$-cycles, and $s_{m}(T)$ be the number of its strongly connected $m$-subtournaments. Due to work of L.W. Beineke and F. Harary, it is well known that $s_{m}(T)\\le s_{m}(RLT_{n}),$ where $RLT_{n}$ is the regular locally transitive tournament of order $n.$ For $m=3$ and $m=4,$ $c_{m}(T)$ equals $s_{m}(T),$ but it is not so for $m\\ge 5.$ As J.W. Moon pointed out in his note in 1966, the problem of determining the maximum of $c_{m}(T)$ seems very difficult in general (i.e. for $m\\ge 5$). In the present paper, based on the Komarov-Mackey formula for $c_{5}(T)$ obtained recently, we prove that $c_{5}(T)\\le (n+1)n(n-1)(n-2)(n-3)/160$ with equality holding iff $T$ is doubly regular. A formula for $s_{5}(T)$ is also deduced. With the use of it, we show that $s_{5}(T)\\le (n+1)n(n-1)(n-3)(11n-47)/1920$ with equality holding iff $T=RLT_{n}$ or $n=7$ and $T$ is regular or $n=5$ and $T$ is strong. It is also proved that for a regular tournament $T$ of (odd) order $n\\ge 9,$ a lower bound $(n+1)n(n-1)(n-3)(17n-59)/3840\\le s_{5}(T)$ holds with equality iff $T$ is doubly regular. These results are compared with the ones recently obtained by the author for $c_{5}(T).$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-28T16:38:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "odd order",
      "doubly regular",
      "regular locally transitive tournament",
      "komarov-mackey formula",
      "equality holding"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}