Complete regular dessins of odd prime power order
Kan Hu, Naer Wang, Roman Nedela

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 -cell embedding of a connected -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.
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