Enumeration of simple complete topological graphs

AbstractA 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 2Θ(n4). We also show that the number of weak isomorphism classes of simple complete topological graphs with n vertices and n4 crossings is at least 2n(logn−O(1)), which improves the estimate of Harborth and Mengersen