{
  "id": "math/0305161",
  "version": "v1",
  "published": "2003-05-12T01:52:04.000Z",
  "updated": "2003-05-12T01:52:04.000Z",
  "title": "Graphs without repeated cycle lengths",
  "authors": [
    "Chunhui Lai"
  ],
  "comment": "5 pages",
  "journal": "Australasian Journal of Combinatorics 27 2003 101-105",
  "categories": [
    "math.CO"
  ],
  "abstract": "In 1975, P. Erd\\\"{o}s proposed the problem of determining the maximum number $f(n)$ of edges in a graph of $n$ vertices in which any two cycles are of different lengths. In this paper, it is proved that $$f(n)\\geq n+36t$$ for $t=1260r+169 (r\\geq 1)$ and $n \\geq 540t^{2}+{175811/2}t+{7989/2}$. Consequently, $\\liminf\\sb {n \\to \\infty} {f(n)-n \\over \\sqrt n} \\geq \\sqrt {2 + {2 \\over 5}}.$ We make the following conjecture: \\par \\bigskip \\noindent{\\bf Conjecture.} $$\\lim_{n \\to \\infty} {f(n)-n\\over \\sqrt n}=\\sqrt {2.4}.$$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2003-05-12T01:52:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C38",
      "05C35"
    ],
    "keywords": [
      "repeated cycle lengths",
      "conjecture",
      "maximum number"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......5161L"
    }
  }
}