Complete regular dessins of odd prime power order

TL;DR

This paper classifies and counts regular dessins with complete bipartite graphs of odd prime power order using group-theoretic methods, advancing understanding of symmetric embeddings in topological graph theory.

Contribution

It provides a complete enumeration and classification of regular dessins with specific bipartite graphs of odd prime power order, a novel application of group theory in this context.

Findings

Enumeration of isomorphism classes of regular dessins

Classification results for dessins with complete bipartite graphs

Application of group-theoretic methods to topological graph embeddings

Abstract

A dessin is a $2$-cell embedding of a connected $2$-coloured bipartite graph into an orientable closed surface. A dessin is regular if its group of colour- and orientation-preserving automorphisms acts regularly on the edges. In this paper we employ group-theoretic method to determine and enumerate the isomorphism classes of regular dessins with the complete bipartite underlying graphs of odd prime power order.