{
  "id": "1608.03501",
  "version": "v1",
  "published": "2016-08-11T15:20:53.000Z",
  "updated": "2016-08-11T15:20:53.000Z",
  "title": "On the relationship between distinguishing number and distinguishing index of a graph",
  "authors": [
    "Saeid Alikhani",
    "Samaneh Soltani"
  ],
  "comment": "5 pages, 2 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling (edge labeling) with $d$ labels that is preserved only by a trivial automorphism. Motivated by a theorem (Kalinowski and Pil\\'sniak, 2015) which state that for every graph of order $n \\geqslant 3$, $D'(G) \\leqslant D(G) + 1$, we characterize all finite simple connected graphs with $D'(G) = D(G) + 1$. Also, we describe all finite simple connected graphs with $D'(G) = D(G)$ and $| E(G)| \\leqslant | V(G)|$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-11T15:20:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15",
      "05E18"
    ],
    "keywords": [
      "distinguishing number",
      "distinguishing index",
      "finite simple connected graphs",
      "relationship",
      "trivial automorphism"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}