Regular embeddings of Kn,n where n is an odd prime power

  • Authors:

  • Gareth A. Jones;Roman Nedela;Martin Škoviera

  • Affiliations:

  • School of Mathematics, University of Southampton, Southampton SO17 1BJ, UK;Institute of Mathematics, Slovak Academy of Science, Severná 5, 975 49 Banská Bystrica, Slovakia;Department of Informatics, Comenius University, 842 48 Bratislava, Slovakia

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

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Abstract

We show that if n=p^e where p is an odd prime and e=1, then the complete bipartite graph K"n","n has p^e^-^1 regular embeddings in orientable surfaces. These maps, which are Cayley maps for cyclic and dihedral groups, have type {2n,n} and genus (n-1)(n-2)/2; one is reflexible, and the rest are chiral. The method involves groups which factorise as a product of two cyclic groups of order n. We deduce that if n is odd then K"n","n has at least n/@?"p"|"np orientable regular embeddings, and that this lower bound is attained if and only if no two primes p and q dividing n satisfy p=1mod(q).