The Rectilinear Crossing Number of K_10 is 62

Abstract

A drawing of a graph G in the plane is said to be a rectilinear drawing of G if the edges are required to be line segments (as opposed to Jordan curves). We assume no three vertices are collinear. The rectilinear crossing number of G is the fewest number of edge crossings attainable over all rectilinear drawings of G. Thanks to Richard Guy, exact values of the rectilinear crossing number of K_n, the complete graph on n vertices, for n = 3,...,9, are known (Guy 1972,