{
  "id": "2409.00645",
  "version": "v1",
  "published": "2024-09-01T07:36:19.000Z",
  "updated": "2024-09-01T07:36:19.000Z",
  "title": "On isomorphisms of $m$-Cayley digraphs",
  "authors": [
    "Xing Zhang",
    "Yuan-Quan Feng",
    "Fu-Gang Yin",
    "Jin-Xin Zhou"
  ],
  "comment": "29",
  "categories": [
    "math.CO"
  ],
  "abstract": "The isomorphism problem for digraphs is a fundamental problem in graph theory. This problem for Cayley digraphs has been extensively investigated over the last half a century. In this paper, we consider this problem for $m$-Cayley digraphs which are generalization of Cayley digraphs. Let $m$ be a positive integer. A digraph admitting a group $G$ of automorphisms acting semiregularly on the vertices with exactly $m$ orbits is called an $m$-Cayley digraph of $G$. In particular, $1$-Cayley digraph is just the Cayley digraph. We first characterize the normalizer of $G$ in the full automorphism group of an $m$-Cayley digraph of a finite group $G$. This generalizes a similar result for Cayley digraph achieved by Godsil in 1981. Then we use this to study the isomorphisms of $m$-Cayley digraphs. The CI-property of a Cayley digraph (CI stands for `Cayley isomorphism') and the DCI-groups (whose Cayley digraphs are all CI-digraphs) are two key topics in the study of isomorphisms of Cayley digraphs. We generalize these concepts into $m$-Cayley digraphs by defining $m$CI- and $m$PCI-digraphs, and correspondingly, $m$DCI- and $m$PDCI-groups. Analogues to Babai's criterion for CI-digraphs are given for $m$CI- and $m$PCI-digraphs, respectively. With these we then classify finite $m$DCI-groups for each $m\\geq 2$, and finite $m$PDCI-groups for each $m\\geq 4$. Similar results are also obtained for $m$-Cayley graphs. Note that 1DCI-groups are just DCI-groups, and the classification of finite DCI-groups is a long-standing open problem that has been worked on a lot.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-01T07:36:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C25",
      "20B25"
    ],
    "keywords": [
      "cayley digraph",
      "similar result",
      "dci-groups",
      "full automorphism group",
      "open problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}