{
  "id": "2204.03630",
  "version": "v1",
  "published": "2022-04-07T17:52:20.000Z",
  "updated": "2022-04-07T17:52:20.000Z",
  "title": "Existence of $2$-Factors in Tough Graphs without Forbidden Subgraphs",
  "authors": [
    "Elizabeth Grimm",
    "Songling Shan",
    "Anna Johnsen"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "For a given graph $R$, a graph $G$ is $R$-free if $G$ does not contain $R$ as an induced subgraph. It is known that every $2$-tough graph with at least three vertices has a $2$-factor. In graphs with restricted structures, it was shown that every $2K_2$-free $3/2$-tough graph with at least three vertices has a $2$-factor, and the toughness bound $3/2$ is best possible. In viewing $2K_2$, the disjoint union of two edges, as a linear forest, in this paper, for any linear forest $R$ on 5, 6, or 7 vertices, we find the sharp toughness bound $t$ such that every $t$-tough $R$-free graph on at least three vertices has a 2-factor.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-07T17:52:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "tough graph",
      "forbidden subgraphs",
      "linear forest",
      "sharp toughness bound",
      "free graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}