Doubly Even Orientable Closed 2-cell Embeddings of the Complete Graph

For all m > 1 and k > 2, we construct closed 2-cell embeddings of the complete graph K8km+4k+1 with faces of size 4k in orientable surfaces. Moreover, we show that when k > 3 there are at least (2m − 1)!/2(2m + 1) = 22mlog2m−O(m) nonisomorphic embeddings of this type. We also show that when k = 2 there are at least 1 4π 1 2m− 5 4 ( 4m e2 ) √ m (1− o(1)) nonisomorphic embeddings of this type.