Genus of a complete bipartite graph

Question

How do I prove that the complete bipartite graph $K_{m,n}$ has genus $\lceil (m−2)(n−2)/4 \rceil$?

I know that the genus of $K_n$ is $\lceil (n-3)(n−4)/12 \rceil$ but I cannot obtain the required proof.

Answer 1

Proven in:

• Ringel, G. "Das Geschlecht des vollständiger Paaren Graphen." Abh. Math. Sem. Univ. Hamburg 28, 139-150, 1965.
• Harary, F. Graph Theory, Reading, MA: Addison-Wesley, 1994, page 119.