{
  "id": "math/0611416",
  "version": "v2",
  "published": "2006-11-14T13:46:48.000Z",
  "updated": "2007-06-21T05:18:42.000Z",
  "title": "Random Graph-Homomorphisms and Logarithmic Degree",
  "authors": [
    "Itai Benjamini",
    "Ariel Yadin",
    "Amir Yehudayoff"
  ],
  "journal": "Electronic Journal of Probability, 12 (2007), 926--950",
  "categories": [
    "math.PR",
    "math-ph",
    "math.CO",
    "math.MP"
  ],
  "abstract": "A graph homomorphism between two graphs is a map from the vertex set of one graph to the vertex set of the other graph, that maps edges to edges. In this note we study the range of a uniformly chosen homomorphism from a graph G to the infinite line Z. It is shown that if the maximal degree of G is `sub-logarithmic', then the range of such a homomorphism is super-constant. Furthermore, some examples are provided, suggesting that perhaps for graphs with super-logarithmic degree, the range of a typical homomorphism is bounded. In particular, a sharp transition is shown for a specific family of graphs C_{n,k} (which is the tensor product of the n-cycle and a complete graph, with self-loops, of size k). That is, given any function psi(n) tending to infinity, the range of a typical homomorphism of C_{n,k} is super-constant for k = 2 log(n) - psi(n), and is 3 for k = 2 log(n) + psi(n).",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-06-21T05:18:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60C05"
    ],
    "keywords": [
      "logarithmic degree",
      "random graph-homomorphisms",
      "vertex set",
      "typical homomorphism",
      "maps edges"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math.....11416B"
    }
  }
}