Classification of nonorientable regular embeddings of complete bipartite graphs

TL;DR

This paper classifies nonorientable regular embeddings of complete bipartite graphs $K_{n,n}$, identifying conditions on $n$ for their existence and counting the number of such embeddings up to isomorphism.

Contribution

It provides a complete classification of nonorientable regular embeddings of $K_{n,n}$ and determines their count based on the prime factorization of $n$.

Findings

Regular embeddings exist only when all prime factors of $n$ are congruent to ±1 mod 8.

Such embeddings exist only for specific $n$ with prime decomposition involving these primes.

Number of embeddings up to isomorphism is $2^k$, where $k$ is the number of prime factors.

Abstract

A 2-cell embedding of a graph $G$ into a closed (orientable or nonorientable) surface is called regular if its automorphism group acts regularly on the flags - mutually incident vertex-edge-face triples. In this paper, we classify the regular embeddings of complete bipartite graphs $K_{n,n}$ into nonorientable surfaces. Such regular embedding of $K_{n,n}$ exists only when $n = 2p_1^{a_1}p_2^{a_2}... p_k^{a_k}$ (a prime decomposition of $n$) and all $p_i \equiv \pm 1 (\mod 8)$. In this case, the number of those regular embeddings of $K_{n,n}$ up to isomorphism is $2^k$.