{
  "id": "2310.10183",
  "version": "v1",
  "published": "2023-10-16T08:45:54.000Z",
  "updated": "2023-10-16T08:45:54.000Z",
  "title": "Toughness and existence of $2$-factors",
  "authors": [
    "Leyou Xu",
    "Bo Zhou"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A graph is $t$-tough if the deletion of any set of, say, $m$ vertices from the graph leaves a graph with at most $\\frac{m}{t}$ components. In 1973, Chv\\'{a}tal suggested the problem of relating toughness to factors in graphs. In 1985, Enomoto et al. showed that each $2$-tough graph with at least three vertices has a $2$-factor, but for any $\\epsilon>0$, there exists a $(2-\\epsilon)$-tough graph on at least $3$ vertices having no $2$-factor. In recent years, the study of sufficient conditions for graphs with toughness less than $2$ having a $2$-factor has received a paramount interest. In this paper, we give new tight sufficient conditions for a $t$-tough graph having a $2$-factor when $1\\le t<2$ by involving independence number, minimum degree, connectivity and forbidden forests.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-16T08:45:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "tough graph",
      "tight sufficient conditions",
      "forbidden forests",
      "paramount interest",
      "minimum degree"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}