{
  "id": "1910.07349",
  "version": "v1",
  "published": "2019-10-16T13:56:24.000Z",
  "updated": "2019-10-16T13:56:24.000Z",
  "title": "On the probability that a random subtree is spanning",
  "authors": [
    "Stephan Wagner"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "We consider the quantity $P(G)$ associated with a graph $G$ that is defined as the probability that a randomly chosen subtree of $G$ is spanning. Motivated by conjectures due to Chin, Gordon, MacPhee and Vincent on the behaviour of this graph invariant depending on the edge density, we establish first that $P(G)$ is bounded below by a positive constant provided that the minimum degree is bounded below by a linear function in the number of vertices. Thereafter, the focus is shifted to the classical Erd\\H{o}s-R\\'enyi random graph model $G(n,p)$. It is shown that $P(G)$ converges in probability to $e^{-1/(ep_{\\infty})}$ if $p \\to p_{\\infty} > 0$ and to $0$ if $p \\to 0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-16T13:56:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C05",
      "05C80"
    ],
    "keywords": [
      "random subtree",
      "probability",
      "random graph model",
      "randomly chosen subtree",
      "linear function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}