{
  "id": "1103.0067",
  "version": "v1",
  "published": "2011-03-01T02:18:36.000Z",
  "updated": "2011-03-01T02:18:36.000Z",
  "title": "Cycle-saturated graphs with minimum number of edges",
  "authors": [
    "Zoltan Furedi",
    "Younjin Kim"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A graph $G$ is called $H$-saturated if it does not contain any copy of $H$, but for any edge $e$ in the complement of $G$ the graph $G+e$ contains some $H$. The minimum size of an $n$-vertex $H$-saturated graph is denoted by $\\sat(n,H)$. We prove $$\\sat(n,C_k) = n + n/k + O((n/k^2) + k^2)$$ holds for all $n\\geq k\\geq 3$, where $C_k$ is a cycle with length $k$. We have a similar result for semi-saturated graphs $$\\ssat(n,C_k) = n + n/(2k) + O((n/k^2) + k).$$ We conjecture that our three constructions are optimal.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-03-01T02:18:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "minimum number",
      "cycle-saturated graphs",
      "similar result",
      "complement",
      "semi-saturated graphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1103.0067F"
    }
  }
}