{
  "id": "1505.01475",
  "version": "v1",
  "published": "2015-05-06T19:43:00.000Z",
  "updated": "2015-05-06T19:43:00.000Z",
  "title": "Which Haar graphs are Cayley graphs?",
  "authors": [
    "István Estélyi",
    "Tomaž Pisanski"
  ],
  "comment": "13 pages, 2 figures",
  "categories": [
    "math.GR",
    "math.CO"
  ],
  "abstract": "For a finite group $G$ and subset $S$ of $G,$ the Haar graph $H(G,S)$ is a bipartite regular graph, defined as a regular $G$-cover of a dipole with $|S|$ parallel arcs labelled by elements of $S$. If $G$ is an abelian group, then $H(G,S)$ is well-known to be a Cayley graph; however, there are examples of non-abelian groups $G$ and subsets $S$ when this is not the case. In this paper we address the problem of classifying finite non-abelian groups $G$ with the property that every Haar graph $H(G,S)$ is a Cayley graph. An equivalent condition for $H(G,S)$ to be a Cayley graph of a group containing $G$ is derived in terms of $G, S$ and $\\mathrm{Aut }G$. It is also shown that the dihedral groups, which are solutions to the above problem, are $\\mathbb{Z}_2^2,D_3,D_4$ and $D_{5}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-05-06T19:43:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20B25",
      "05C25",
      "05E10"
    ],
    "keywords": [
      "cayley graph",
      "haar graph",
      "bipartite regular graph",
      "classifying finite non-abelian groups",
      "parallel arcs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}