{
  "id": "1509.05857",
  "version": "v1",
  "published": "2015-09-19T08:06:46.000Z",
  "updated": "2015-09-19T08:06:46.000Z",
  "title": "Cliques in C_4-free graphs of large minimum degree",
  "authors": [
    "A. Gyarfas",
    "G. N. Sarkozy"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A graph $G$ is called $C_4$-free if it does not contain the cycle $C_4$ as an induced subgraph. Hubenko, Solymosi and the first author proved (answering a question of Erd\\H os) a peculiar property of $C_4$-free graphs: $C_4$ graphs with $n$ vertices and average degree at least $cn$ contain a complete subgraph (clique) of size at least $c'n$ (with $c'= 0.1c^2n$). We prove here better bounds (${c^2n\\over 2+c}$ in general and $(c-1/3)n$ when $ c \\le 0.733$) from the stronger assumption that the $C_4$-free graphs have minimum degree at least $cn$. Our main result is a theorem for regular graphs, conjectured in the paper mentioned above: $2k$-regular $C_4$-free graphs on $4k+1$ vertices contain a clique of size $k+1$. This is best possible shown by the $k$-th power of the cycle $C_{4k+1}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-09-19T08:06:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C35",
      "05C75"
    ],
    "keywords": [
      "large minimum degree",
      "free graphs",
      "first author",
      "average degree",
      "complete subgraph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150905857G"
    }
  }
}