{
  "id": "2107.01416",
  "version": "v1",
  "published": "2021-07-03T11:59:42.000Z",
  "updated": "2021-07-03T11:59:42.000Z",
  "title": "The maximum size of a graph with prescribed order, circumference and minimum degree",
  "authors": [
    "Leilei Zhang"
  ],
  "comment": "11 pages, 0 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Erd\\H{o}s determined the maximum size of a nonhamiltonian graph of order $n$ and minimum degree at least $k$ in 1962. Recently, Ning and Peng generalized Erd\\H{o}s' work and gave the maximum size $\\psi(n,c,k)$ of graphs with prescribed order $n$, circumference $c$ and minimum degree at least $k.$ But for some triples $n,c,k,$ the maximum size is not attained by a graph of minimum degree $k.$ For example, $\\psi(15,14,3)=77$ is attained by a unique graph of minimum degree $7,$ not $3.$ In this paper we obtain more precise information by determining the maximum size of a graph with prescribed order, circumference and minimum degree. Consequently we solve the corresponding problem for longest paths. All these results on the size of graphs have clique versions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-03T11:59:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C30",
      "05C35",
      "05C38"
    ],
    "keywords": [
      "minimum degree",
      "maximum size",
      "prescribed order",
      "circumference",
      "nonhamiltonian graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}