{
  "id": "2310.13190",
  "version": "v1",
  "published": "2023-10-19T22:59:43.000Z",
  "updated": "2023-10-19T22:59:43.000Z",
  "title": "A hypergraph analog of Dirac's Theorem for long cycles in 2-connected graphs, II: Large uniformities",
  "authors": [
    "Alexandr Kostochka",
    "Ruth Luo",
    "Grace McCourt"
  ],
  "comment": "23 pages, 1 figure. arXiv admin note: text overlap with arXiv:2212.14516",
  "categories": [
    "math.CO"
  ],
  "abstract": "Dirac proved that each $n$-vertex $2$-connected graph with minimum degree $k$ contains a cycle of length at least $\\min\\{2k, n\\}$. We obtain analogous results for Berge cycles in hypergraphs. Recently, the authors proved an exact lower bound on the minimum degree ensuring a Berge cycle of length at least $\\min\\{2k, n\\}$ in $n$-vertex $r$-uniform $2$-connected hypergraphs when $k \\geq r+2$. In this paper we address the case $k \\leq r+1$ in which the bounds have a different behavior. We prove that each $n$-vertex $r$-uniform $2$-connected hypergraph $H$ with minimum degree $k$ contains a Berge cycle of length at least $\\min\\{2k,n,|E(H)|\\}$. If $|E(H)|\\geq n$, this bound coincides with the bound of the Dirac's Theorem for 2-connected graphs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-19T22:59:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D05",
      "05C65",
      "05C38",
      "05C35"
    ],
    "keywords": [
      "diracs theorem",
      "hypergraph analog",
      "long cycles",
      "large uniformities",
      "berge cycle"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}