{
  "id": "1807.01116",
  "version": "v1",
  "published": "2018-07-03T12:29:05.000Z",
  "updated": "2018-07-03T12:29:05.000Z",
  "title": "On symmetries of edge and vertex colourings of graphs",
  "authors": [
    "Florian Lehner",
    "Simon M. Smith"
  ],
  "comment": "13 Pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $c$ and $c'$ be edge or vertex colourings of a graph $G$. We say that $c'$ is less symmetric than $c$ if the stabiliser (in $\\operatorname{Aut} G$) of $c'$ is contained in the stabiliser of $c$. We show that if $G$ is not a bicentred tree, then for every vertex colouring of $G$ there is a less symmetric edge colouring with the same number of colours. On the other hand, if $T$ is a tree, then for every edge colouring there is a less symmetric vertex colouring with the same number of edges. Our results can be used to characterise those graphs whose distinguishing index is larger than their distinguishing number.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-03T12:29:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "symmetries",
      "stabiliser",
      "symmetric edge colouring",
      "bicentred tree",
      "symmetric vertex colouring"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}