{
  "id": "1306.1803",
  "version": "v1",
  "published": "2013-06-07T18:29:51.000Z",
  "updated": "2013-06-07T18:29:51.000Z",
  "title": "The maximum number of complete subgraphs in a graph with given maximum degree",
  "authors": [
    "Jonathan Cutler",
    "A. J. Radcliffe"
  ],
  "comment": "14 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Extremal problems involving the enumeration of graph substructures have a long history in graph theory. For example, the number of independent sets in a $d$-regular graph on $n$ vertices is at most $(2^{d+1}-1)^{n/2d}$ by the Kahn-Zhao theorem. Relaxing the regularity constraint to a minimum degree condition, Galvin conjectured that, for $n\\geq 2d$, the number of independent sets in a graph with $\\delta(G)\\geq d$ is at most that in $K_{d,n-d}$. In this paper, we give a lower bound on the number of independent sets in a $d$-regular graph mirroring the upper bound in the Kahn-Zhao theorem. The main result of this paper is a proof of a strengthened form of Galvin's conjecture, covering the case $n\\leq 2d$ as well. We find it convenient to address this problem from the perspective of $\\complement{G}$. In other words, we give an upper bound on the number of complete subgraphs of a graph $G$ on $n$ vertices with $\\Delta(G)\\leq r$, valid for all values of $n$ and $r$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-06-07T18:29:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C30",
      "05C35"
    ],
    "keywords": [
      "complete subgraphs",
      "maximum degree",
      "maximum number",
      "independent sets",
      "kahn-zhao theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1306.1803C"
    }
  }
}