Summary: We calculate here both exact and asymptotic formulas for the number of regions enclosed by the edges of a regular drawing of the complete bipartite graph $K_{n,n}$. This extends the work of Legendre, who considered the number of crossings arising from such a graph. We also show that the ratio of regions to crossings tends to a rational constant as $n$ increases without limit.

05C62, 05A15, 05A16, 11A05

complete bipartite graph, crossing, greatest common divisor, region