Korzhik proved that there were (2s)! nonisomorphic orientable quadrangular embeddings of \(K_{8s+5}\) , \(s\geqslant 1\) . Hartsfield and Ringel proved that the complete graph \(K_{8s+5}\) had an orientable quadrangular embedding, and the polyhedron determined by this embedding was a minimal quadrangulation of the surface. In this paper, we first construct current graphs of the complete graph \(K_{8ms+4m+1}\) (m and s are natural numbers), and find \([(2s)!]^{2m-1}\) current assignments of the current graph of \(K_{8ms+4m+1}\) . And then, we prove that each current graph of \(K_{8ms+4m+1}\) has at least \([2^{2m-1}\times (2m-1)!]^{2s+1}\) orientable surface embeddings with one face. By these results, we prove that the complete graph \(K_{8ms+4m+1}\) has at least \([2^{2m-1}\times (2m-1)!]^{2s+1}\times [(2s)!]^{2m-1}/2\) nonisomorphic orientable 4m-gonal embeddings, and these embeddings have minimal number of faces. These results include those of of Hartsfield, Ringel and Korzhik, and these results are broader and stronger.
