{
  "id": "1905.00239",
  "version": "v1",
  "published": "2019-05-01T09:52:56.000Z",
  "updated": "2019-05-01T09:52:56.000Z",
  "title": "An Ore-type condition for existence of two disjoint cycles",
  "authors": [
    "Maoqun Wang",
    "Jianguo Qian"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $n_{1}$ and $n_{2}$ be two integers with $n_{1},n_{2}\\geq3$ and $G$ a graph of order $n=n_{1}+n_{2}$. As a generalization of Ore's degree condition for the existence of Hamilton cycle in $G$, El-Zahar proved that if $\\delta(G)\\geq \\left\\lceil\\frac{n_{1}}{2}\\right\\rceil+\\left\\lceil\\frac{n_{2}}{2}\\right\\rceil$ then $G$ contains two disjoint cycles of length $n_{1}$ and $n_{2}$. Recently, Yan et. al considered the problem by extending the degree condition to degree sum condition and proved that if $d(u)+d(v)\\geq n+4$ for any pair of non-adjacent vertices $u$ and $v$ of $G$, then $G$ contains two disjoint cycles of length $n_{1}$ and $n_{2}$. They further asked whether the degree sum condition can be improved to $d(u)+d(v)\\geq n+2$. In this paper, we give a positive answer to this question. Our result also generalizes El-Zahar's result when $n_{1}$ and $n_{2}$ are both odd.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-01T09:52:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C07",
      "05C38"
    ],
    "keywords": [
      "disjoint cycles",
      "ore-type condition",
      "degree sum condition",
      "ores degree condition",
      "generalizes el-zahars result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}