{
  "id": "1701.07472",
  "version": "v1",
  "published": "2017-01-25T20:21:34.000Z",
  "updated": "2017-01-25T20:21:34.000Z",
  "title": "The maximum number of cliques in graphs without long cycles",
  "authors": [
    "Ruth Luo"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "The Erd\\H{o}s--Gallai Theorem states that for $k\\geq 3$ every graph on $n$ vertices with more than $\\frac{1}{2}(k-1)(n-1)$ edges contains a cycle of length at least $k$. Kopylov proved a strengthening of this result for 2-connected graphs with extremal examples $H_{n,k,t}$ and $H_{n,k,2}$. In this note, we generalize the result of Kopylov to bound the number of $s$-cliques in a graph with circumference less than $k$. Furthermore, we show that the same extremal examples that maximize the number of edges also maximize the number of cliques of any fixed size. Finally, we obtain the extremal number of $s$-cliques in a graph with no path on $k$-vertices.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-25T20:21:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "long cycles",
      "maximum number",
      "extremal examples",
      "extremal number",
      "theorem states"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}