{
  "id": "2105.12168",
  "version": "v1",
  "published": "2021-05-25T18:46:23.000Z",
  "updated": "2021-05-25T18:46:23.000Z",
  "title": "The jump of the clique chromatic number of random graphs",
  "authors": [
    "Lyuben Lichev",
    "Dieter Mitsche",
    "Lutz Warnke"
  ],
  "comment": "14 pages",
  "categories": [
    "math.CO",
    "cs.DM",
    "math.PR"
  ],
  "abstract": "The clique chromatic number of a graph is the smallest number of colors in a vertex coloring so that no maximal clique is monochromatic. In 2016 McDiarmid, Mitsche and Pralat noted that around p \\approx n^{-1/2} the clique chromatic number of the random graph G_{n,p} changes by n^{\\Omega(1)} when we increase the edge-probability p by n^{o(1)}, but left the details of this surprising phenomenon as an open problem. We settle this problem, i.e., resolve the nature of this polynomial `jump' of the clique chromatic number of the random graph G_{n,p} around edge-probability p \\approx n^{-1/2}. Our proof uses a mix of approximation and concentration arguments, which enables us to (i) go beyond Janson's inequality used in previous work and (ii) determine the clique chromatic number of G_{n,p} up to logarithmic factors for any edge-probability p.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-25T18:46:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15",
      "05C80",
      "60C05"
    ],
    "keywords": [
      "clique chromatic number",
      "random graph",
      "edge-probability",
      "logarithmic factors",
      "smallest number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}