Enumeration of simple complete topological graphs

A simple topological graph T = ( V ( T ) , E ( T ) ) is a drawing of a graph in the plane, where every two edges have at most one common point (an end-point or a crossing) and no three edges pass through a single crossing. Topological graphs G and H are isomorphic if H can be obtained from G by a homeomorphism of the sphere, and weakly isomorphic if G and H have the same set of pairs of crossing edges. We prove that the number of isomorphism classes of simple complete topological graphs on n vertices is 2 Θ ( n 4 ) . We also show that the number weak isomorphism classes of simple complete topological graphs with n vertices and ( n 4 ) crossings is at least 2 n ( log n − O ( 1 ) ) , which improves the estimate of Harborth an Mengersen.