Show that $K_{r, s}$ is planar if and only if $\min$ {r, s} ≤ 2.

Question

So I've done some draws and this is true, but How can I argument to prove that, by the maximum number of edges in $K$ ?

Or by $d(v)$

Any help?

Answer 1

This basically boils down to a special case of Kuratowski's theorem. In the special case of bipartite graphs, this will say that your graph will be planar if and only if it does not contain $K_{3, 3}$ as a minor. First, you would show that $K_{3, 3}$ is not planar (and hence neither is any bipartite graph containing $K_{3, 3}$ as a minor), and then show that $K_{r, 2}$ is in fact planar for any $r$ (you can just explicitly draw $K_{r, 2}$ as a plane graph in this case to demonstrate planarity).