On the Number of Nonisomorphic Orientable Regular Embeddings of Complete Graphs

In this paper we consider those 2-cell orientable embeddings of a complete graph Kn+1 which are generated by rotation schemes on an abelian group ? of order n+1, where a rotation scheme an ? is defined as a cyclic permutation (s1, s2, ?, sn) of all nonzero elements of ?. It is shown that two orientable embeddings of Kn+1 generated by schemes (s1, s2, ?, sn) and (?1, ?2, ?, ?n) are isomorphic if and only if (?1, ?2, ?, ?n)=(?(s1), ?(s2), ?, ?(sn)) or (?1, ?2, ?, ?n)=(?(sn), ?, ?(s2), ?(s1)), where ? is an automorphism of ?. As a consequence, by representing schemes by index one current graphs, the following results are obtained. The graphs K12s+4 and K12s+7 for every s?1 have at least 4s non-isomorphic face 3-colorable orientable triangular embeddings. The graph K8s+5 for every s?1 has at least 8×16s?1 nonisomorphic orientable quadrangular embeddings.