Regular embeddings of Kn,n where n is a power of 2. I: Metacyclic case

  • Authors:

  • Shao-Fei Du;Gareth Jones;Jin Ho Kwak;Roman Nedela;Martin Škoviera

  • Affiliations:

  • Department of Mathematics, Capital Normal University, Beijing 100037, People's Republic of China;School of Mathematics, University of Southampton, Southampton S017 1BJ, United Kingdom;Combinatorial and Computational Mathematics Center, Pohang University of Science and Technology, Pohang 790-784, Republic of Korea;Mathematical Institute, Slovak Academy of Sciences, Severná 5, 975 49 Banská Bystrica, Slovakia;Faculty of Mathematics, Physics and Informatics, Comenius University, 842 48 Bratislava, Slovakia

  • Venue:
  • European Journal of Combinatorics
  • Year:
  • 2007

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Abstract

A 2-cell embedding of a graph in an orientable closed surface is called regular if its automorphism group acts regularly on arcs of the embedded graph. The aim of this and of the associated consecutive paper is to give a classification of regular embeddings of complete bipartite graphs K"n","n, where n=2^e. The method involves groups G which factorize as a product XY of two cyclic groups of order n so that the two cyclic factors are transposed by an involutory automorphism. In particular, we give a classification of such groups G. Employing the classification we investigate automorphisms of these groups, resulting in a classification of regular embeddings of K"n","n based on that for G. We prove that given n=2^e (for e=3), there are, up to map isomorphism, exactly 2^e^-^2+4 regular embeddings of K"n","n. Our analysis splits naturally into two cases depending on whether the group G is metacyclic or not.