{
  "abstract": "A dessin is a 2-cell embedding of a connected bipartite graph into an orientable closed surface. It is regular if the colour- and orientation-preserving automorphism group acts regularly on edges, and circular if every face boundary is a simple cycle. We classify and enumerate circular regular dessins of the complete bipartite multigraph ${\\bf K}_{2^e,2^f}^{(2^d)}$ ($e\\ge f\\ge 2$, $d\\ge 1$) whose automorphism group is {\\it metacyclic}. This extends the work of Fan (circular, metacyclic) and Cai--Hu (regular, metacyclic) from simple graphs ${\\bf K}_{2^e,2^f}$ to multigraphs ${\\bf K}_{2^e,2^f}^{(2^d)}$. Such dessins correspond to non-exact bicyclic factorisations $G=ła\\rłb\\r\\cong\\ZZ_{2^{e+d}}\\ZZ_{2^{f+d}}$, where $ła\\r\\capłb\\r\\cong\\ZZ_{2^d}$ and $łba\\r\\cap ła\\r=łba\\r\\cap łb\\r=1$. In general, a group admitting such a factorisation need not be metacyclic; in this paper we impose the additional condition that $G$ is a metacyclic 2-group. Under this assumption, we obtain explicit enumeration formulas for the number of such dessins up to isomorphism. Together with the known classifications for simple graphs (the case $d=0$), this completes the enumeration of the metacyclic case for 2-power complete bipartite multigraphs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.41123",
  "author_total": 2,
  "authors": [
    {
      "name": "Yuntian Wu"
    },
    {
      "name": "Wenwen Fan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/er473sswe5qnc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.41123",
  "fulltext_url": null,
  "id": "er473sswe5qnc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.41123",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.41123",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/er473sswe5qnc.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.41123",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.41123"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Circular regular dessins with underlying graphs ${\\bf K}_{2^e,2^f}^{(2^d)}$: the metacyclic case",
  "title_text": "Circular regular dessins with underlying graphs bf K_2^e,2^f^(2^d): the metacyclic case",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/er473sswe5qnc",
  "venue": "arXiv",
  "versions": []
}