Circular regular dessins with underlying graphs bf K2e,2f(2d): the metacyclic case
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 K2e,2f(2d) (e≥ f≥ 2, d≥ 1) whose automorphism group is {\it metacyclic}. This extends the work of Fan (circular, metacyclic) and Cai--Hu (regular, metacyclic) from simple graphs bf K2e,2f to multigraphs bf K2e,2f(2d). Such dessins correspond to non-exact bicyclic factorisations G=łarłbr≅ZZ2e+dZZ2f+d, where łar∩łbr≅ZZ2d and łbar∩ łar=łbar∩ łbr=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.
Related
- Complete regular dessins and non-metacyclic bicyclic 2-groups with cyclic commutator subgroup
- Circular flows and bisections of random regular graphs
- Complete regular dessins: the nonmetacyclic case
- Freeness of Arrangements with Regular Underlying Matroids
- Cayley graphs of abelian groups, circular colorings and discrete fundamental groups
BibTeX
@misc{wu2026circular,
title = {{Circular regular dessins with underlying graphs $\bf K_2^e,2^f^(2^d)$: the metacyclic case}},
author = {Yuntian Wu and Wenwen Fan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41123},
eprint = {2610.41123},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41123},
}
RIS
TY - UNPB
TI - Circular regular dessins with underlying graphs bf K_2^e,2^f^(2^d): the metacyclic case
AU - Yuntian Wu
AU - Wenwen Fan
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41123
UR - https://arxiv.org/abs/2610.41123
AB - 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≥ f≥ 2, d≥ 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=łarłbr≅ZZ_2^e+dZZ_2^f+d, where łar∩łbr≅ZZ_2^d and łbar∩ łar=łbar∩ łbr=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.
ER -