{
  "id": "2212.01781",
  "version": "v1",
  "published": "2022-12-04T10:04:36.000Z",
  "updated": "2022-12-04T10:04:36.000Z",
  "title": "A note on regular sets in Cayley graphs",
  "authors": [
    "Junyang Zhang",
    "Yanhong Zhu"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A subset $R$ of the vertex set of a graph $\\Gamma$ is said to be $(\\kappa,\\tau)$-regular if $R$ induces a $\\kappa$-regular subgraph and every vertex outside $R$ is adjacent to exactly $\\tau$ vertices in $R$. In particular, if $R$ is a $(\\kappa,\\tau)$-regular set of some Cayley graph on a finite group $G$, then $R$ is called a $(\\kappa,\\tau)$-regular set of $G$. Let $H$ be a non-trivial normal subgroup of $G$, and $\\kappa$ and $\\tau$ a pair of integers satisfying $0\\leq\\kappa\\leq|H|-1$, $1\\leq\\tau\\leq|H|$ and $\\gcd(2,|H|-1)\\mid\\kappa$. It is proved that (i) if $\\tau$ is even, then $H$ is a $(\\kappa,\\tau)$-regular set of $G$; (ii) if $\\tau$ is odd, then $H$ is a $(\\kappa,\\tau)$-regular set of $G$ if and only if it is a $(0,1)$-regular set of $G$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-04T10:04:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "regular set",
      "cayley graph",
      "non-trivial normal subgroup",
      "finite group",
      "vertex outside"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}