{
  "id": "0912.2919",
  "version": "v2",
  "published": "2009-12-15T15:10:31.000Z",
  "updated": "2011-05-25T20:10:13.000Z",
  "title": "Toughness and Vertex Degrees",
  "authors": [
    "D. Bauer",
    "H. J. Broersma",
    "J. van den Heuvel",
    "N. Kahl",
    "E. Schmeichel"
  ],
  "comment": "13 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "We study theorems giving sufficient conditions on the vertex degrees of a graph $G$ to guarantee $G$ is $t$-tough. We first give a best monotone theorem when $t\\ge1$, but then show that for any integer $k\\ge1$, a best monotone theorem for $t=\\frac1k\\le 1$ requires at least $f(k)\\cdot|V(G)|$ nonredundant conditions, where $f(k)$ grows superpolynomially as $k\\rightarrow\\infty$. When $t<1$, we give an additional, simple theorem for $G$ to be $t$-tough, in terms of its vertex degrees.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-05-25T20:10:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C40",
      "05C07"
    ],
    "keywords": [
      "vertex degrees",
      "best monotone theorem",
      "study theorems giving sufficient conditions",
      "simple theorem",
      "nonredundant conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0912.2919B"
    }
  }
}