{
  "id": "1708.01211",
  "version": "v1",
  "published": "2017-08-03T16:56:51.000Z",
  "updated": "2017-08-03T16:56:51.000Z",
  "title": "A Ramsey Property of Random Regular and $k$-out Graphs",
  "authors": [
    "Michael Anastos",
    "Deepak Bal"
  ],
  "comment": "8 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "In this note we consider a Ramsey property of random $d$-regular graphs, $\\mathcal{G}(n,d)$. Let $r\\ge 2$ be fixed. Then w.h.p. the edges of $\\mathcal{G}(n, 2r)$ can be colored such that every monochromatic component has size $o(n)$. On the other hand, there exists a constant $\\gamma > 0$ such that w.h.p., every $r$-coloring of the edges of $\\mathcal{G}(n, 2r+1)$ must contain a monochromatic cycle of length at least $\\gamma n$. We prove an analogous result for random $k$-out graphs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-03T16:56:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "ramsey property",
      "random regular",
      "monochromatic cycle",
      "regular graphs",
      "monochromatic component"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}