{
  "id": "2006.09697",
  "version": "v1",
  "published": "2020-06-17T07:51:16.000Z",
  "updated": "2020-06-17T07:51:16.000Z",
  "title": "Longest and shortest cycles in random planar graphs",
  "authors": [
    "Mihyun Kang",
    "Michael Missethan"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $P(n,m)$ be a graph chosen uniformly at random from the class of all planar graphs on vertex set $\\{1, \\ldots, n\\}$ with $m=m(n)$ edges. We study the cycle and block structure of $P(n,m)$ when $m\\sim n/2$. More precisely, we determine the asymptotic order of the length of the longest and shortest cycle in $P(n,m)$ in the critical range when $m=n/2+o(n)$. In addition, we describe the block structure of $P(n,m)$ in the weakly supercritical regime when $n^{2/3}\\ll m-n/2\\ll n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-17T07:51:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random planar graphs",
      "shortest cycle",
      "block structure",
      "vertex set",
      "asymptotic order"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}