The thickness of a simple graph G is the smallest number of planar subgraphs of G that have G as their union.
Show that K3,3 has 2 as its thickness.
My attempt:
[If we decompose K(3,3) into any pair of nonempty graphs A and B, the union of their edges make the edges of K(3,3), and A and B have no edges in common), then both A and B are planar. (Example: let A be one arbitrary edge of K(3,3) and let B be the rest.) It follows that the thickness of K(3,3) is no more than 2.]
It is enough? how to prove it more complete