{
  "id": "1406.7552",
  "version": "v1",
  "published": "2014-06-29T20:04:02.000Z",
  "updated": "2014-06-29T20:04:02.000Z",
  "title": "Highly linked tournaments",
  "authors": [
    "Alexey Pokrovskiy"
  ],
  "comment": "8 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "A (possibly directed) graph is $k$-linked if for any two disjoint sets of vertices $\\{x_1, \\dots, x_k\\}$ and $\\{y_1, \\dots, y_k\\}$ there are vertex disjoint paths $P_1, \\dots, P_k$ such that $P_i$ goes from $x_i$ to $y_{i}$. A theorem of Bollob\\'as and Thomason says that every $22k$-connected (undirected) graph is $k$-linked. It is desirable to obtain analogues for directed graphs as well. Although Thomassen showed that the Bollob\\'as-Thomason Theorem does not hold for general directed graphs, he proved an analogue of the theorem for tournaments - there is a function $f(k)$ such that every strongly $f(k)$-connected tournament is $k$-linked. The bound on $f(k)$ was reduced to $O(k \\log k)$ by K\\\"uhn, Lapinskas, Osthus, and Patel, who also conjectured that a linear bound should hold. We prove this conjecture, by showing that every strongly $452k$-connected tournament is $k$-linked.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-06-29T20:04:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C40",
      "05C20"
    ],
    "keywords": [
      "highly linked tournaments",
      "connected tournament",
      "vertex disjoint paths",
      "conjecture",
      "disjoint sets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1406.7552P"
    }
  }
}