{
  "id": "1111.5629",
  "version": "v3",
  "published": "2011-11-23T21:21:56.000Z",
  "updated": "2012-05-21T00:08:00.000Z",
  "title": "An improved upper bound for the bondage number of graphs on surfaces",
  "authors": [
    "Jia Huang"
  ],
  "comment": "8 pages, to appear in Discrete Mathematics",
  "journal": "Discrete Mathematics 312 (2012) 2776-2781",
  "categories": [
    "math.CO"
  ],
  "abstract": "The bondage number $b(G)$ of a graph $G$ is the smallest number of edges whose removal from $G$ results in a graph with larger domination number. Recently Gagarin and Zverovich showed that, for a graph $G$ with maximum degree $\\Delta(G)$ and embeddable on an orientable surface of genus $h$ and a non-orientable surface of genus $k$, $b(G)\\leq\\min\\{\\Delta(G)+h+2,\\Delta+k+1\\}$. They also gave examples showing that adjustments of their proofs implicitly provide better results for larger values of $h$ and $k$. In this paper we establish an improved explicit upper bound for $b(G)$, using the Euler characteristic $\\chi$ instead of the genera $h$ and $k$, with the relations $\\chi=2-2h$ and $\\chi=2-k$. We show that $b(G)\\leq\\Delta(G)+\\lfloor r\\rfloor$ for the case $\\chi\\leq0$ (i.e. $h\\geq1$ or $k\\geq2$), where $r$ is the largest real root of the cubic equation $z^3+2z^2+(6\\chi-7)z+18\\chi-24=0$. Our proof is based on the technique developed by Carlson-Develin and Gagarin-Zverovich, and includes some elementary calculus as a new ingredient. We also find an asymptotically equivalent result $b(G)\\leq\\Delta(G)+\\lceil\\sqrt{12-6\\chi\\,}-1/2\\rceil$ for $\\chi\\leq0$, and a further improvement for graphs with large girth.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-05-21T00:08:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C69"
    ],
    "keywords": [
      "bondage number",
      "larger domination number",
      "explicit upper bound",
      "largest real root",
      "smallest number"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1111.5629H"
    }
  }
}