{
  "id": "2011.11605",
  "version": "v1",
  "published": "2020-11-23T18:24:38.000Z",
  "updated": "2020-11-23T18:24:38.000Z",
  "title": "Disjoint cycles with length constraints in digraphs of large connectivity or minimum degree",
  "authors": [
    "Raphael Steiner"
  ],
  "comment": "20 pages, 4 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "A conjecture by Lichiardopol states that for every $k \\ge 1$ there exists an integer $g(k)$ such that every digraph of minimum out-degree at least $g(k)$ contains $k$ vertex-disjoint directed cycles of pairwise distinct lengths. Motivated by Lichiardopol's conjecture, we study the existence of vertex-disjoint directed cycles satisfying length constraints in digraphs of large connectivity or large minimum degree. Our main result is that for every $k \\in \\mathbb{N}$, there exists $s(k) \\in \\mathbb{N}$ such that every strongly $s(k)$-connected digraph contains $k$ vertex-disjoint directed cycles of pairwise distinct lengths. In contrast, for every $k \\in \\mathbb{N}$ we construct a strongly $k$-connected digraph containing no two vertex- or arc-disjoint directed cycles of the same length. It is an open problem whether $g(3)$ exists. Here we prove the existence of an integer $K$ such that every digraph of minimum out- and in-degree at least $K$ contains $3$ vertex-disjoint directed cycles of pairwise distinct lengths.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-23T18:24:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C07",
      "05C20",
      "05C38",
      "05C40"
    ],
    "keywords": [
      "vertex-disjoint directed cycles",
      "large connectivity",
      "minimum degree",
      "pairwise distinct lengths",
      "disjoint cycles"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}