{
  "id": "2006.11104",
  "version": "v1",
  "published": "2020-06-19T12:40:26.000Z",
  "updated": "2020-06-19T12:40:26.000Z",
  "title": "On subgroup perfect codes in Cayley graphs",
  "authors": [
    "Junyang Zhang",
    "Sanming Zhou"
  ],
  "comment": "Final version, European Journal of Combinatorics, to appear",
  "categories": [
    "math.CO"
  ],
  "abstract": "A perfect code in a graph $\\Gamma = (V, E)$ is a subset $C$ of $V$ such that no two vertices in $C$ are adjacent and every vertex in $V \\setminus C$ is adjacent to exactly one vertex in $C$. A subgroup $H$ of a group $G$ is called a subgroup perfect code of $G$ if there exists a Cayley graph of $G$ which admits $H$ as a perfect code. Equivalently, $H$ is a subgroup perfect code of $G$ if there exists an inverse-closed subset $A$ of $G$ containing the identity element such that $(A, H)$ is a tiling of $G$ in the sense that every element of $G$ can be uniquely expressed as the product of an element of $A$ and an element of $H$. In this paper we obtain multiple results on subgroup perfect codes of finite groups, including a few necessary and sufficient conditions for a subgroup of a finite group to be a subgroup perfect code, a few results involving $2$-subgroups in the study of subgroup perfect codes, and several results on subgroup perfect codes of metabelian groups, generalized dihedral groups, nilpotent groups and $2$-groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-19T12:40:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C25",
      "05C69",
      "94B25"
    ],
    "keywords": [
      "subgroup perfect code",
      "cayley graph",
      "finite group",
      "generalized dihedral groups",
      "metabelian groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}