{
  "id": "2003.09750",
  "version": "v1",
  "published": "2020-03-21T20:55:45.000Z",
  "updated": "2020-03-21T20:55:45.000Z",
  "title": "Large cycles in essentially 4-connected graphs",
  "authors": [
    "Michael Wigal",
    "Xingxing Yu"
  ],
  "comment": "17 pages, 4 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Tutte proved that every 4-connected planar graph contains a Hamilton cycle, but there are 3-connected $n$-vertex planar graphs whose longest cycles have length $\\Theta(n^{\\log_32})$. On the other hand, Jackson and Wormald in 1992 proved that an essentially 4-connected $n$-vertex planar graph contains a cycle of length at least $(2n+4)/5$, which was recently improved to $5(n+2)/8$ by Fabrici {\\it et al}. In this paper, we improve this bound to $\\lceil (2n+6)/3\\rceil$ for $n\\ge 6$, which is best possible, by proving a quantitative version of a result of Thomassen on Tutte paths.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-21T20:55:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C38",
      "05C40",
      "05C45"
    ],
    "keywords": [
      "large cycles",
      "vertex planar graph contains",
      "longest cycles",
      "tutte paths",
      "hamilton cycle"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}