{
  "id": "2208.06851",
  "version": "v1",
  "published": "2022-08-14T13:30:30.000Z",
  "updated": "2022-08-14T13:30:30.000Z",
  "title": "An improved lower bound on the length of the longest cycle in random graphs",
  "authors": [
    "Michael Anastos"
  ],
  "categories": [
    "math.CO",
    "cs.DS"
  ],
  "abstract": "We provide a new lower bound on the length of the longest cycle of the binomial random graph $G(n,(1+\\epsilon)/n)$ that holds w.h.p. for all $\\epsilon=\\epsilon(n)$ such that $\\epsilon^3n\\to \\infty$. In the case $\\epsilon\\leq \\epsilon_0$ for some sufficiently small constant $\\epsilon_0$, this bound is equal to $1.581\\epsilon^2n$ which improves upon the current best lower bound of $4\\epsilon^2n/3$ due to Luczak.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-14T13:30:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C80",
      "05C38"
    ],
    "keywords": [
      "longest cycle",
      "current best lower bound",
      "binomial random graph",
      "sufficiently small constant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}