A Parity Theorem for Drawings of Complete and Complete Bipartite Graphs

Forty years ago, Kleitman considered the numbers of crossings in good planar drawings of the complete bipartite graph ${K_{m,n}}$. Among other things, he proved that, for $m$ and $n$ both odd, the parities of these numbers of crossings are all the same. His proof was sufficiently controversial that he provided another proof a few years later. In this work, we provide a complete, simple proof based on counting, elementary graph theory, and the Jordan Curve Theorem.