{
  "id": "2308.11434",
  "version": "v1",
  "published": "2023-08-22T13:35:33.000Z",
  "updated": "2023-08-22T13:35:33.000Z",
  "title": "On the subgroup regular set in Cayley graphs",
  "authors": [
    "Asamin Khaefi",
    "Zeinab Akhlaghi",
    "Behrooz Khosravi"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A subset $C$ of the vertex set of a graph $\\Gamma$ is said to be $(a,b)$-regular if $C$ induces an $a$-regular subgraph and every vertex outside $C$ is adjacent to exactly $b$ vertices in $C$. In particular, if $C$ is an $(a,b)$-regular set of some Cayley graph on a finite group $G$, then $C$ is called an $(a,b)$-regular set of $G$ and a $(0,1)$-regular set is called a perfect code of $G$. In [Wang, Xia and Zhou, Regular sets in Cayley graphs, J. Algebr. Comb., 2022] it is proved that if $H$ is a normal subgroup of $G$, then $H$ is a perfect code of $G$ if and only if it is an $(a,b)$-regular set of $G$, for each $0\\leq a\\leq|H|-1$ and $0\\leq b\\leq|H|$ with $\\gcd(2,|H|-1)\\mid a$. In this paper, we generalize this result and show that a subgroup $H$ of $G$ is a perfect code of $G$ if and only if it is an $(a,b)$-regular set of $G$, for each $0\\leq a\\leq|H|-1$ and $0\\leq b\\leq|H|$ such that $\\gcd(2,|H|-1)$ divides $a$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-22T13:35:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "cayley graph",
      "subgroup regular set",
      "perfect code",
      "vertex set",
      "regular subgraph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}