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 phi of Order n + 1, where a rotation scheme an phi is defined as a cyclic permu tation (beta (1,) beta (2)..... beta (n), of all nonzero elements of phi. I t is shown that two orientable embeddings of Kn+1 generated by schemes (bet a (1), beta (2),...beta (n)) and (gamma (1),gamma (2),....,gamma (n)) are i somorphic if and only if (gamma (1),gamma (2),...,gamma (n)) = (phi(beta (1 )), phi(beta (2)),...,phi(beta (n))) or (gamma (1),gamma (2),...gamma (n)) = (phi(beta (n)),....phi(beta (2)), phi(beta (1))). where phi is an automor phism of phi. As a consequence. by representing schemes by index one curren t graphs, the following results are obtained. The graphs K12s + 4 and K12s + 7 of every s greater than or equal to 1 have at least 4(s) nonisomorphic isomorphic face 3-colorable orientable triangular embedding. The graph K8s + 5 for every s greater than or equal to 1 have at least 8 x 16(s-1) noniso morphic orientable quadrangular embeddings. (C) 2001 Academic Press.