{
  "id": "1010.2516",
  "version": "v2",
  "published": "2010-10-12T20:49:48.000Z",
  "updated": "2011-05-25T22:08:46.000Z",
  "title": "Asymptotic enumeration of sparse 2-connected graphs",
  "authors": [
    "Graeme Kemkes",
    "Cristiane M. Sato",
    "Nicholas Wormald"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "We determine an asymptotic formula for the number of labelled 2-connected (simple) graphs on $n$ vertices and $m$ edges, provided that $m-n\\to\\infty$ and $m=O(n\\log n)$ as $n\\to\\infty$. This is the entire range of $m$ not covered by previous results. The proof involves determining properties of the core and kernel of random graphs with minimum degree at least 2. The case of 2-edge-connectedness is treated similarly. We also obtain formulae for the number of 2-connected graphs with given degree sequence for most (`typical') sequences. Our main result solves a problem of Wright from 1983.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-05-25T22:08:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "asymptotic enumeration",
      "entire range",
      "asymptotic formula",
      "minimum degree",
      "degree sequence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1010.2516K"
    }
  }
}