{
  "id": "1702.02662",
  "version": "v1",
  "published": "2017-02-09T00:18:52.000Z",
  "updated": "2017-02-09T00:18:52.000Z",
  "title": "The maximum number of cycles in a graph with fixed number of edges",
  "authors": [
    "Andrii Arman",
    "Sergei Tsaturian"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "The main topic considered is maximizing the number of cycles in a graph with given number of edges. In 2009, Kir\\'{a}li conjectured that there is constant $c$ such that any graph with $m$ edges has at most $(1.4)^m$ cycles. In this paper, it is shown that for sufficiently large $m$, a graph with $m$ edges has at most $(1.443)^m$ cycles. For sufficiently large $m$, examples of a graph with $m$ edges and $(1.37)^m$ cycles are presented. For a graph with given number of vertices and edges an upper bound on the maximal number of cycles is given. Also, exponentially tight bounds are proved for the maximum number of cycles in a multigraph with given number of edges, as well as in a multigraph with given number of vertices and edges.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-02-09T00:18:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "maximum number",
      "fixed number",
      "sufficiently large",
      "main topic",
      "exponentially tight bounds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}