{
  "id": "1703.01874",
  "version": "v1",
  "published": "2017-03-06T14:02:43.000Z",
  "updated": "2017-03-06T14:02:43.000Z",
  "title": "Distinguishing number and distinguishing index of strong product of two graphs",
  "authors": [
    "Samaneh Soltani",
    "Saeid Alikhani"
  ],
  "comment": "6 pages",
  "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. The strong product $G\\boxtimes H$ of two graphs $G$ and $H$ is the graph with vertex set $V (G)\\times V (H)$ and edge set $\\{\\{(x_1, x_2), (y_1, y_2)\\} | x_iy_i \\in E(G_i) ~{\\rm or}~ x_i = y_i ~{\\rm for~ each}~ 1 \\leq i \\leq 2.\\}$. In this paper we study the distinguishing number and the distinguishing index of strong product of two graphs. We prove that for every $k \\geq 2$, the $k$-th strong power of a connected $S$-thin graph $G$ has distinguishing index equal 2.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-06T14:02:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15",
      "05C60"
    ],
    "keywords": [
      "strong product",
      "distinguishing number",
      "th strong power",
      "vertex set",
      "edge set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}