{
  "id": "2009.03032",
  "version": "v1",
  "published": "2020-09-07T11:57:52.000Z",
  "updated": "2020-09-07T11:57:52.000Z",
  "title": "A Degree Condition for a Graph to Have All $(a,b)$-Factors",
  "authors": [
    "Haodong Liu",
    "Hongliang Lu"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $a$ and $b$ be positive integers such that $a\\leq b$ and $a\\equiv b\\pmod 2$. We say that $G$ has all $(a, b)$-parity factors if $G$ has an $h$-factor for every function $h: V(G) \\rightarrow \\{a,a+2,\\ldots,b-2,b\\}$ with $b|V(G)|$ even and $h(v)\\equiv b\\pmod 2$ for all $v\\in V(G)$. In this paper, we prove that every graph $G$ with $n\\geq 3(b+1)(a+b)$ vertices has all $(a,b)$-parity factors if $\\delta(G)\\geq (b^2-b)/a$, and for any two nonadjacent vertices $u,v \\in V(G)$, $\\max\\{d_G(u),d_G(v)\\}\\geq \\frac{bn}{a+b}$. Moreover, we show that this result is best possible in some sense.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-07T11:57:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "degree condition",
      "parity factors",
      "nonadjacent vertices",
      "positive integers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}