{
  "id": "2308.14778",
  "version": "v1",
  "published": "2023-08-28T13:34:50.000Z",
  "updated": "2023-08-28T13:34:50.000Z",
  "title": "A precise condition for independent transversals in bipartite covers",
  "authors": [
    "Stijn Cambie",
    "Penny Haxell",
    "Ross J. Kang",
    "Ronen Wdowinski"
  ],
  "comment": "10 pages, 2 figures",
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "abstract": "Given a bipartite graph $H=(V=V_A\\cup V_B,E)$ in which any vertex in $V_A$ (resp. $V_B$) has degree at most $D_A$ (resp. $D_B$), suppose there is a partition of $V$ that is a refinement of the bipartition $V_A\\cup V_B$ such that the parts in $V_A$ (resp. $V_B$) have size at least $k_A$ (resp. $k_B$). We prove that the condition $D_A/k_A+D_B/k_B\\le 1$ is sufficient for the existence of an independent set of vertices of $H$ that is simultaneously transversal to the partition, and show moreover that this condition is sharp. This result is a bipartite refinement of two well-known results on independent transversals, one due to the second author the other due to Szab\\'o and Tardos.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-28T13:34:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C69",
      "05D15",
      "05C15",
      "05C35"
    ],
    "keywords": [
      "independent transversals",
      "bipartite covers",
      "precise condition",
      "independent set",
      "bipartite graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}