{
  "id": "2307.13196",
  "version": "v1",
  "published": "2023-07-25T01:24:25.000Z",
  "updated": "2023-07-25T01:24:25.000Z",
  "title": "An Infinite Family of Connected 1-Factorisations of Complete 3-Uniform Hypergraphs",
  "authors": [
    "Barbara Maenhaut",
    "Jeremy Mitchell",
    "Anna Puskas"
  ],
  "comment": "11 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "A connected 1-factorisation is a 1-factorisation of a hypergraph for which the union of each pair of distinct 1-factors is a connected hypergraph. A uniform 1-factorisation is a 1-factorisation of a hypergraph for which the union of each pair of distinct 1-factors is isomorphic to the same subhypergraph, and a uniform-connected 1-factorisation is a uniform 1-factorisation in which that subhypergraph is connected. Chen and Lu [Journal of Algebraic Combinatorics, 46(2) 475--497, 2017] describe a family of 1-factorisations of the complete 3-uniform hypergraph on $q+1$ vertices, where $q\\equiv 2\\pmod 3$ is a prime power. In this paper, we show that their construction yields a connected 1-factorisation only when $q=2,5,11$ or $q=2^p$ for some odd prime $p$, and a uniform 1-factorisation only for $q=2,5,8$ (each of these is a uniform-connected 1-factorisation).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-25T01:24:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C51",
      "05E18"
    ],
    "keywords": [
      "infinite family",
      "construction yields",
      "odd prime",
      "prime power",
      "subhypergraph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}