{
  "id": "2006.13459",
  "version": "v1",
  "published": "2020-06-24T04:04:39.000Z",
  "updated": "2020-06-24T04:04:39.000Z",
  "title": "Connected cubic graphs with the maximum number of perfect matchings",
  "authors": [
    "Peter Horak",
    "Dongryul Kim"
  ],
  "comment": "20 pages, 19 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "It is proved that for $n \\geq 6$, the number of perfect matchings in a simple connected cubic graph on $2n$ vertices is at most $4 f_{n-1}$, with $f_n$ being the $n$-th Fibonacci number. The unique extremal graph is characterized as well. In addition, it is shown that the number of perfect matchings in any cubic graph $G$ equals the expected value of a random variable defined on all $2$-colorings of edges of $G$. Finally, an improved lower bound on the maximum number of cycles in a cubic graph is provided.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-24T04:04:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C35"
    ],
    "keywords": [
      "perfect matchings",
      "maximum number",
      "th fibonacci number",
      "unique extremal graph",
      "simple connected cubic graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}