{
  "id": "2108.02639",
  "version": "v1",
  "published": "2021-08-05T14:25:28.000Z",
  "updated": "2021-08-05T14:25:28.000Z",
  "title": "Every $(13k-6)$-strong tournament with minimum out-degree at least $28k-13$ is $k$-linked",
  "authors": [
    "Jørgen Bang-Jensen",
    "Kasper Skov Johansen"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A digraph $D$ is $k$-linked if it satisfies that for every choice of disjoint sets $\\{x_1,\\ldots{},x_k\\}$ and $\\{y_1,\\ldots{},y_k\\}$ of vertices of $D$ there are vertex disjoint paths $P_1,\\ldots{},P_k$ such that $P_i$ is an $(x_i,y_i)$-path. Confirming a conjecture by K\\\"uhn et al, Pokrovskiy proved in 2015 that every $452k$-strong tournament is $k$-linked and asked for a better linear bound. Very recently Meng et al proved that every $(40k-31)$-strong tournament is $k$-linked. In this note we use an important lemma from their paper to give a short proof that every $(13k-6)$-strong tournament of minimum out-degree at least $28k-13$ is $k$-linked.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-05T14:25:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C20",
      "05C40"
    ],
    "keywords": [
      "strong tournament",
      "minimum out-degree",
      "vertex disjoint paths",
      "better linear bound",
      "disjoint sets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}