{
  "id": "2003.02107",
  "version": "v1",
  "published": "2020-03-04T14:40:49.000Z",
  "updated": "2020-03-04T14:40:49.000Z",
  "title": "Arc-disjoint in- and out-branchings in digraphs of independence number at most 2",
  "authors": [
    "Joergen Bang-Jensen",
    "Stephane Bessy",
    "Frederic Havet",
    "Anders Yeo"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "We prove that every digraph of independence number at most 2 and arc-connectivity at least 2 has an out-branching $B^+$ and an in-branching $B^-$ which are arc-disjoint (we call such branchings good pair). This is best possible in terms of the arc-connectivity as there are infinitely many strong digraphs with independence number 2 and arbitrarily high minimum in-and out-degrees that have good no pair. The result settles a conjecture by Thomassen for digraphs of independence number 2. We prove that every digraph on at most 6 vertices and arc-connectivity at least 2 has a good pair and give an example of a 2-arc-strong digraph $D$ on 10 vertices with independence number 4 that has no good pair. We also show that there are infinitely many digraphs with independence number 7 and arc-connectivity 2 that have no good pair. Finally we pose a number of open problems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-04T14:40:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "independence number",
      "arc-disjoint",
      "arc-connectivity",
      "arbitrarily high minimum in-and out-degrees",
      "out-branching"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}