{
  "id": "2102.00323",
  "version": "v1",
  "published": "2021-01-30T22:04:17.000Z",
  "updated": "2021-01-30T22:04:17.000Z",
  "title": "Paths of Length Three are $K_{r+1}$-Turán Good",
  "authors": [
    "Kyle Murphy",
    "JD Nir"
  ],
  "comment": "24 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "The generalized Tur\\'an problem $ext(n,T,F)$ is to determine the maximal number of copies of a graph $T$ that can exist in an $F$-free graph on $n$ vertices. Recently, Gerbner and Palmer noted that the solution to the generalized Tur\\'an problem is often the original Tur\\'an graph. They gave the name \"$F$-Tur\\'an-good\" to graphs $T$ for which, for large enough $n$, the solution to the generalized Tur\\'an problem is realized by a Tur\\'an graph. They prove that the path graph on two edges, $P_2$, is $K_{r+1}$-Tur\\'an-good for all $r \\ge 3$, but they conjecture that the same result should hold for all $P_\\ell$. In this paper, using arguments based in flag algebras, we prove that the path on three edges, $P_3$, is also $K_{r+1}$-Tur\\'an-good for all $r \\ge 3$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-30T22:04:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "generalized turan problem",
      "turan-good",
      "original turan graph",
      "flag algebras",
      "free graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}