{
  "id": "2006.05100",
  "version": "v1",
  "published": "2020-06-09T08:02:48.000Z",
  "updated": "2020-06-09T08:02:48.000Z",
  "title": "Perfect sets in Cayley graphs",
  "authors": [
    "Yanpeng Wang",
    "Binzhou Xia",
    "Sanming Zhou"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "In a graph $\\Gamma$ with vertex set $V$, a subset $C$ of $V$ is called an $(a,b)$-perfect set if every vertex in $C$ has exactly $a$ neighbors in $C$ and every vertex in $V\\setminus C$ has exactly $b$ neighbors in $C$, where $a$ and $b$ are nonnegative integers. In the literature $(0,1)$-perfect sets are known as perfect codes and $(1,1)$-perfect sets are known as total perfect codes. In this paper we prove that, for any finite group $G$, if a non-trivial normal subgroup $H$ of $G$ is a perfect code in some Cayley graph of $G$, then it is also an $(a,b)$-perfect set in some Cayley graph of $G$ for any pair of integers $a$ and $b$ with $0\\leqslant a\\leqslant|H|-1$ and $0\\leqslant b\\leqslant |H|$ such that $\\gcd(2,|H|-1)$ divides $a$. A similar result involving total perfect codes is also proved in the paper.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-09T08:02:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C25",
      "05E18",
      "94B25"
    ],
    "keywords": [
      "perfect set",
      "cayley graph",
      "total perfect codes",
      "non-trivial normal subgroup",
      "finite group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}