Regular dessins with underlying graphs [formula omitted]: The metacyclic case

Abstract

A dessin is a 2-cell embedding of a connected 2-colored bipartite graph into an orientable closed surface. A dessin is regular if its group of automorphisms, preserving both color and orientation, acts transitively on its edge set. Leveraging the theory of metacyclic 2-groups, we undertake the classification and enumeration of regular dessins with complete bipartite underlying graphs of the form K2e,2f, while imposing the additional criterion of metacyclic automorphism groups. Our findings extend and generalize the earlier work of Du et al. (2007) [5].