{
  "id": "1409.0972",
  "version": "v1",
  "published": "2014-09-03T07:14:37.000Z",
  "updated": "2014-09-03T07:14:37.000Z",
  "title": "Cycles in Oriented 3-graphs",
  "authors": [
    "Imre Leader",
    "Ta Sheng Tan"
  ],
  "comment": "12 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "An oriented 3-graph consists of a family of triples (3-sets), each of which is given one of its two possible cyclic orientations. A cycle in an oriented 3-graph is a positive sum of some of the triples that gives weight zero to each 2-set. Our aim in this paper is to consider the following question: how large can the girth of an oriented 3-graph (on $n$ vertices) be? We show that there exist oriented 3-graphs whose shortest cycle has length $\\frac{n^2}{2}(1+o(1))$: this is asymptotically best possible. We also show that there exist 3-tournaments whose shortest cycle has length $\\frac{n^2}{3}(1+o(1))$, in complete contrast to the case of 2-tournaments.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-09-03T07:14:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "shortest cycle",
      "cyclic orientations",
      "weight zero",
      "complete contrast",
      "positive sum"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1409.0972L"
    }
  }
}