{
  "id": "2205.10874",
  "version": "v1",
  "published": "2022-05-22T17:28:02.000Z",
  "updated": "2022-05-22T17:28:02.000Z",
  "title": "Toughness and the existence of tree-connected $\\{f,f+k\\}$-factors",
  "authors": [
    "Morteza Hasanvand"
  ],
  "comment": "This paper is an improved version of a removed part of the paper arXiv:1702.06203. arXiv admin note: substantial text overlap with arXiv:2205.05044",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G$ be a graph and let $f$ be a positive integer-valued function on $V(G)$ satisfying $2m\\le f\\le b$, where $b$ and $m$ are two positive integers with $b\\ge 4m^2$. In this paper, we show that if $G$ is $b^2$-tough and $|V(G)|\\ge b^2$, then it has an $m$-tree-connected factor $H$ such that for each vertex $v$, $$d_H(v)\\in \\{f(v), f(v)+1\\}.$$ Next, we generalize this result by giving sufficient conditions for a tough graph to have a tree-connected factors $H$ such that for each vertex $v$, $d_H(v)\\in \\{f(v), f(v)+k\\}$. As an application, we prove that every $64b(b-a)^2$-tough graph $G$ of order at least $b+1$ with $ab|V(G)|$ even admits a connected factor whose degrees lie in the set $\\{a,b\\}$, where $a$ and $b$ are two integers with $2\\le a< b < \\frac{6}{5}a$. Moreover, we prove that every $16$-tough graph $G$ of order at least three admits a $2$-connected factor whose degrees lie in the set $\\{2,3\\}$, provided that $G$ has a $2$-factor with girth at least five. This result confirms a weaker version of a long-standing conjecture due to Chv\\'atal (1973).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-22T17:28:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "tough graph",
      "degrees lie",
      "tree-connected factor",
      "giving sufficient conditions",
      "result confirms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}