{
  "id": "2001.05905",
  "version": "v1",
  "published": "2020-01-16T15:50:44.000Z",
  "updated": "2020-01-16T15:50:44.000Z",
  "title": "Almost-2-regular random graphs",
  "authors": [
    "Lorenzo Federico"
  ],
  "comment": "16 pages, 1 figure",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "We study a special case of the configuration model, in which almost all the vertices of the graph have degree $2$. We show that the graph has a very peculiar and interesting behaviour, in particular when the graph is made up by a vast majority of vertices of degree $2$ and a vanishing proportion of vertices of higher degree, the giant component contains $n(1-o(1))$ vertices, but the second component can still grow polynomially in $n$. On the other hand, when almost all the vertices have degree $2$ except for $o(n)$ which have degree $1$, there is no component of linear size.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-16T15:50:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C38",
      "05C80",
      "60C05"
    ],
    "keywords": [
      "random graphs",
      "giant component contains",
      "configuration model",
      "special case",
      "vast majority"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}