Abstract: Let C_m,n be the join graph of C_m(a cycle of ength m) and n isolated vertices. In this paper, we first show hat the genus and nonorientable genus of C_m,n equal those of K_m,n, which were well known and discovered by Ringel [1, 2].
Then we show that the complete bipartite graph K_m,n has a strong orientable genus embedding if m≥2 and n≥2 and has a strong nonorientable genus embedding if m≥3 and n≥3.

Key words: nonorientable genus, strong embedding, genus, nonorientable genus, strong embedding