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 Φ of order n+1, where a rotation scheme an Φ is defined as a cyclic permutation (β1, β2, …, βn) of all nonzero elements of Φ. It is shown that two orientable embeddings of Kn+1 generated by schemes (β1, β2, …, βn) and (γ1, γ2, …, γn) are isomorphic if and only if (γ1, γ2, …, γn)=(ϕ(β1), ϕ(β2), …, ϕ(βn)) or (γ1, γ2, …, γn)=(ϕ(βn), …, ϕ(β2), ϕ(β1)), where ϕ is an automorphism of Φ. 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.