{
  "id": "1803.03240",
  "version": "v1",
  "published": "2018-03-08T18:27:32.000Z",
  "updated": "2018-03-08T18:27:32.000Z",
  "title": "The maximum number of $P_\\ell$ copies in $P_k$-free graphs",
  "authors": [
    "Ervin Győri",
    "Nika Salia",
    "Casey Tompkins",
    "Oscar Zamora"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Generalizing Tur\\'an's classical extremal problem, Alon and Shikhelman investigated the problem of maximizing the number of $T$ copies in an $H$-free graph, for a pair of graphs $T$ and $H$. Whereas Alon and Shikhelman were primarily interested in determining the order of magnitude for large classes of graphs $H$, we focus on the case when $T$ and $H$ are paths, where we find asymptotic and in some cases exact results. We also consider other structures like stars and the set of cycles of length at least $k$, where we derive asymptotically sharp estimates. Our results generalize well-known extremal theorems of Erd\\H{o}s and Gallai.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-08T18:27:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "free graph",
      "maximum number",
      "results generalize well-known extremal theorems",
      "cases exact results",
      "generalizing turans classical extremal problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}