{
  "id": "math/0406542",
  "version": "v2",
  "published": "2004-06-26T16:47:26.000Z",
  "updated": "2005-03-17T18:39:53.000Z",
  "title": "Distinguishing numbers for graphs and groups",
  "authors": [
    "Julianna S. Tymoczko"
  ],
  "comment": "13 pages; final version",
  "journal": "Electronic Journal of Combinatorics, 11 (1) (2004), #R63",
  "categories": [
    "math.CO",
    "math.GR"
  ],
  "abstract": "A graph G is distinguished if its vertices are labelled by a map \\phi: V(G) \\longrightarrow {1,2,...,k} so that no graph automorphism preserves \\phi. The distinguishing number of G is the minimum number k necessary for \\phi to distinguish the graph. It is one measure of the complexity of the graph. We extend these definitions to an arbitrary group action of G on a set X. A labelling \\phi: X \\longrightarrow {1,2,...,k} is distinguishing if no nontrivial element of G preserves \\phi except those in the stabilizer of X. The distinguishing number of the group action on X is the minimum k needed for \\phi to distinguish the group action. We show that distinguishing group actions is a more general problem than distinguishing graphs. We completely characterize actions of the symmetric group S_n on a set with distinguishing number n.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2005-03-17T18:39:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15",
      "05C25",
      "20D60"
    ],
    "keywords": [
      "distinguishing number",
      "graph automorphism preserves",
      "arbitrary group action",
      "minimum number",
      "nontrivial element"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......6542T"
    }
  }
}