Regular embeddings ofKn,nwherenis an odd prime power

Abstract

We show that if n=pe where p is an odd prime and e≥1, then the complete bipartite graph Kn,n has pe−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 Kn,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).