Regular embeddings of complete bipartite graphs: classification and enumeration

TL;DR

This paper classifies and counts all regular embeddings of complete bipartite graphs in orientable surfaces, analyzing their automorphism groups and combinatorial features using group theory and previous classifications.

Contribution

It provides a comprehensive classification and enumeration of regular embeddings of $K_{n,n}$, extending prior work for prime power cases and applying advanced group theory techniques.

Findings

Complete classification of regular embeddings of $K_{n,n}$

Enumeration formulas for these embeddings

Analysis of automorphism groups and combinatorial properties

Abstract

The regular embeddings of complete bipartite graphs $K_{n,n}$ in orientable surfaces are classified and enumerated, and their automorphism groups and combinatorial properties are determined. The method depends on earlier classifications in the cases where $n$ is a prime power, obtained in collaboration with Du, Kwak, Nedela and \v{S}koviera, together with results of It\^o, Hall, Huppert and Wielandt on factorisable groups and on finite solvable groups.