{
  "id": "1704.04674",
  "version": "v1",
  "published": "2017-04-15T18:54:50.000Z",
  "updated": "2017-04-15T18:54:50.000Z",
  "title": "Limit Theorems for Monochromatic 2-Stars and Triangles",
  "authors": [
    "Bhaswar B. Bhattacharya",
    "Sumit Mukherjee"
  ],
  "comment": "35 pages, 6 figures",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "In this paper we study the asymptotic limiting distribution of the number of monochromatic 2-stars and triangles in a growing sequence of graphs where the vertices are colored independently and uniformly at random. We show that the asymptotic distribution of the number of monochromatic 2-stars for any graph sequence is a sum of mutually independent components, each term of which is a polynomial of a single Poisson random variable of degree 1 or 2. Moreover, any limiting distribution for the number of monochromatic 2-stars has a representation of this form. The number of monochromatic triangles is more challenging to characterize; in this case, the limiting distributions can involve products and mixtures of Poisson random variables. We derive conditions under which the number of monochromatic triangles is asymptotically Poisson. As a consequence, an interesting second-moment phenomenon emerges: the number of monochromatic 2-stars or triangles converges to a Poisson distribution whenever the limits of their corresponding mean and the variance are equal.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-15T18:54:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15",
      "60C05",
      "60F05",
      "05D99"
    ],
    "keywords": [
      "limit theorems",
      "monochromatic triangles",
      "interesting second-moment phenomenon emerges",
      "asymptotic limiting distribution",
      "mutually independent components"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}