Computational search of small point sets with small rectilinear crossing number - Fabila-Monroy, Ruy; López, Jorge (Journal of Graph Algorithms and Applications, 2014)
Let cr(Kn) be the minimum number of crossings over all rectilinear drawings of the complete graph on n vertices in the plane. In this paper we prove that cr(Kn) < 0.380473(n4)+Θ(n3); improving thus on the previous best known upper bound. This is done by obtaining new rectilinear drawings of Kn for small values of n, and then using known constructions to obtain arbitrarily large good drawings from smaller ones. The "small" sets were found using a simple heuristic detailed in this paper.