{
  "id": "2011.07612",
  "version": "v1",
  "published": "2020-11-15T19:36:42.000Z",
  "updated": "2020-11-15T19:36:42.000Z",
  "title": "Triangles in randomly perturbed graphs",
  "authors": [
    "Julia Böttcher",
    "Olaf Parczyk",
    "Amedeo Sgueglia",
    "Jozef Skokan"
  ],
  "comment": "30 pages, 1 figure",
  "categories": [
    "math.CO"
  ],
  "abstract": "We study the problem of finding pairwise vertex-disjoint triangles in the randomly perturbed graph model, which is the union of any $n$-vertex graph $G$ with linear minimum degree and the binomial random graph $G(n,p)$. We prove that asymptotically almost surely $G \\cup G(n,p)$ contains $\\min \\{\\delta(G), \\lfloor n/3 \\rfloor \\}$ pairwise vertex-disjoint triangles, provided $p \\ge C \\log n/n$, where $C$ is a large enough constant. This is a perturbed version of an old result of Dirac. Our result is asymptotically optimal and answers a question of Han, Morris, and Treglown [RSA, to appear] in the case of triangle-factors. Together with a result of Balogh, Treglown, and Wagner [CPC, 2019, no. 2, 159-176] this fully resolves the existence of triangle-factors in randomly perturbed graphs. We also prove a stability version of our result. Finally, we discuss further generalisations to larger clique-factors, larger cycle-factors, and $2$-universality.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-15T19:36:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "binomial random graph",
      "linear minimum degree",
      "vertex graph",
      "finding pairwise vertex-disjoint triangles",
      "randomly perturbed graph model"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}