{
  "id": "2501.10723",
  "version": "v1",
  "published": "2025-01-18T10:46:38.000Z",
  "updated": "2025-01-18T10:46:38.000Z",
  "title": "Cyclic $m$-DCI-groups and $m$-CI-groups",
  "authors": [
    "István Kovács",
    "Luka Šinkovec"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Based on the earlier work of Li (European J. Combin. 1997) and Dobson (Discrete Math. 2008), in this paper we complete the classification of cyclic $m$-DCI-groups and $m$-CI-groups. For a positive integer $m$ such that $m \\ge 3$, we show that the group $\\mathbb{Z}_n$ is an $m$-DCI-group if and only if $n$ is not divisible by $8$ nor by $p^2$ for any odd prime $p < m$. Furthermore, if $m \\ge 6$, then we show that $\\mathbb{Z}_n$ is an $m$-CI-group if and only if either $n \\in \\{ 8, 9, 18 \\}$, or $n \\notin \\{ 8, 9, 18 \\}$ and $n$ is not divisible by $8$ nor by $p^2$ for any odd prime $p < \\frac{m - 1}{2}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-01-18T10:46:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C25",
      "20B25"
    ],
    "keywords": [
      "odd prime",
      "discrete math",
      "earlier work",
      "classification",
      "positive integer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}