{
  "id": "1407.4845",
  "version": "v1",
  "published": "2014-07-17T21:24:17.000Z",
  "updated": "2014-07-17T21:24:17.000Z",
  "title": "Hamiltonicity and $σ$-hypergraphs",
  "authors": [
    "Christina Zarb"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "We define and study a special type of hypergraph. A $\\sigma$-hypergraph $H= H(n,r,q$ $\\mid$ $\\sigma$), where $\\sigma$ is a partition of $r$, is an $r$-uniform hypergraph having $nq$ vertices partitioned into $ n$ classes of $q$ vertices each. If the classes are denoted by $V_1$, $V_2$,...,$V_n$, then a subset $K$ of $V(H)$ of size $r$ is an edge if the partition of $r$ formed by the non-zero cardinalities $ \\mid$ $K$ $\\cap$ $V_i \\mid$, $ 1 \\leq i \\leq n$, is $\\sigma$. The non-empty intersections $K$ $\\cap$ $V_i$ are called the parts of $K$, and $s(\\sigma)$ denotes the number of parts. We consider various types of cycles in hypergraphs such as Berge cycles and sharp cycles in which only consecutive edges have a nonempty intersection. We show that most $\\sigma$-hypergraphs contain a Hamiltonian Berge cycle and that, for $n \\geq s+1$ and $q \\geq r(r-1)$, a $\\sigma$-hypergraph $H$ always contains a sharp Hamiltonian cycle. We also extend this result to $k$-intersecting cycles.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-17T21:24:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C65",
      "05C45"
    ],
    "keywords": [
      "hamiltonicity",
      "hamiltonian berge cycle",
      "sharp hamiltonian cycle",
      "non-zero cardinalities",
      "non-empty intersections"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.4845Z"
    }
  }
}