{
  "id": "1504.02650",
  "version": "v1",
  "published": "2015-04-10T12:09:48.000Z",
  "updated": "2015-04-10T12:09:48.000Z",
  "title": "Transversals in $4$-Uniform Hypergraphs",
  "authors": [
    "Michael A. Henning",
    "Anders Yeo"
  ],
  "comment": "41 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $H$ be a $3$-regular $4$-uniform hypergraph on $n$ vertices. The transversal number $\\tau(H)$ of $H$ is the minimum number of vertices that intersect every edge. Lai and Chang [J. Combin. Theory Ser. B 50 (1990), 129--133] proved that $\\tau(H) \\le 7n/18$. Thomass\\'{e} and Yeo [Combinatorica 27 (2007), 473--487] improved this bound and showed that $\\tau(H) \\le 8n/21$. We provide a further improvement and prove that $\\tau(H) \\le 3n/8$, which is best possible due to a hypergraph of order eight. More generally, we show that if $H$ is a $4$-uniform hypergraph on $n$ vertices and $m$ edges with maximum degree $\\Delta(H) \\le 3$, then $\\tau(H) \\le n/4 + m/6$, which proves a known conjecture. We show that an easy corollary of our main result is that the total domination number of a graph on $n$ vertices with minimum degree at least~4 is at most $3n/7$, which was the main result of the Thomass\\'{e}-Yeo paper [Combinatorica 27 (2007), 473--487].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-04-10T12:09:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C65"
    ],
    "keywords": [
      "uniform hypergraph",
      "main result",
      "total domination number",
      "minimum number",
      "transversal number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150402650H"
    }
  }
}