{
  "id": "2002.00765",
  "version": "v1",
  "published": "2020-01-30T23:11:59.000Z",
  "updated": "2020-01-30T23:11:59.000Z",
  "title": "New upper bounds for the bondage number of a graph in terms of its maximum degree and Euler characteristic",
  "authors": [
    "Jia Huang",
    "Jian Shen"
  ],
  "comment": "11 pages. arXiv admin note: text overlap with arXiv:1111.5629",
  "journal": "Ars Combinatoria 140 (2018) 373--387",
  "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. Let $G$ be embeddable on a surface whose Euler characteristic $\\chi$ is as large as possible, and assume $\\chi\\leq0$. Gagarin-Zverovich and Huang have recently found upper bounds of $b(G)$ in terms of the maximum degree $\\Delta(G)$ and the Euler characteristic $\\chi(G)=\\chi$. In this paper we prove a better upper bound $b(G)\\leq\\Delta(G)+\\lfloor t\\rfloor$ where $t$ is the largest real root of the cubic equation $z^3 + z^2 + (3\\chi - 8)z + 9\\chi - 12=0$; this upper bound is asymptotically equivalent to $b(G)\\leq\\Delta(G)+1+\\lfloor \\sqrt{4-3\\chi} \\rfloor$. We also establish further improved upper bounds for $b(G)$ when the girth, order, or size of the graph $G$ is large compared with its Euler characteristic $\\chi$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-30T23:11:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "euler characteristic",
      "maximum degree",
      "bondage number",
      "largest real root",
      "larger domination number"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}