{
  "id": "1405.6560",
  "version": "v1",
  "published": "2014-05-26T13:02:03.000Z",
  "updated": "2014-05-26T13:02:03.000Z",
  "title": "Sharp threshold for embedding combs and other spanning trees in random graphs",
  "authors": [
    "Richard Montgomery"
  ],
  "comment": "20 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "When $k|n$, the tree $\\mathrm{Comb}_{n,k}$ consists of a path containing $n/k$ vertices, each of whose vertices has a disjoint path length $k-1$ beginning at it. We show that, for any $k=k(n)$ and $\\epsilon>0$, the binomial random graph $\\mathcal{G}(n,(1+\\epsilon)\\log n/ n)$ almost surely contains $\\mathrm{Comb}_{n,k}$ as a subgraph. This improves a recent result of Kahn, Lubetzky and Wormald. We prove a similar statement for a more general class of trees containing both these combs and all bounded degree spanning trees which have at least $\\epsilon n/ \\log^9n$ disjoint bare paths length $\\lceil\\log^9 n\\rceil$. We also give an efficient method for finding large expander subgraphs in a binomial random graph. This allows us to improve a result on almost spanning trees by Balogh, Csaba, Pei and Samotij.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-26T13:02:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C05",
      "05C80"
    ],
    "keywords": [
      "spanning trees",
      "sharp threshold",
      "embedding combs",
      "binomial random graph",
      "disjoint bare paths length"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.6560M"
    }
  }
}