{
  "id": "1605.04585",
  "version": "v1",
  "published": "2016-05-15T18:03:56.000Z",
  "updated": "2016-05-15T18:03:56.000Z",
  "title": "Small subgraphs in the trace of a random walk",
  "authors": [
    "Michael Krivelevich",
    "Peleg Michaeli"
  ],
  "comment": "17 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "We consider the combinatorial properties of the trace of a random walk on the complete graph and on the random graph $G(n,p)$. In particular, we study the appearance of a fixed subgraph in the trace. We prove that for a subgraph containing a cycle, the threshold for its appearance in the trace of a random walk of length $m$ is essentially equal to the threshold for its appearance in the random graph drawn from $G(n,m)$. In the case where the base graph is the complete graph, we show that a fixed forest appears in the trace typically much earlier than it appears in $G(n,m)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-05-15T18:03:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C81",
      "05C80",
      "60G50"
    ],
    "keywords": [
      "random walk",
      "small subgraphs",
      "complete graph",
      "appearance",
      "random graph drawn"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}