Classification of nonorientable regular embeddings of complete bipartite graphs

2010/01/18 by Jin Ho Kwak, Kwak, Jin Ho, Young Soo Kwon +1
Mathematics · #05C10 #05C30 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05C10 #msc:05C30

paper · pdf · doi:10.48550/arxiv.1001.2936

arxiv created 2010/01/18 · arxiv updated 2010/02/26

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 = 2p₁^a₁p₂^a₂... p_k^a_k (a prime decomposition of n) and all p_i ≡ ± 1 (\mod 8). In this case, the number of those regular embeddings of K_n,n up to isomorphism is 2^k.

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