Exponentially many nonisomorphic genus embeddings of

We prove that for every n , m ≥ 6 , the complete bipartite graph K n , m has at least 1 8 n m 2 ⌊ ( n − 1 ) / 3 ⌋ ⌊ ( m − 2 ) / 4 ⌋ nonisomorphic orientable as well as nonorientable genus embeddings.