Toward the rectilinear crossing number of Kn: new drawings, upper bounds, and asymptotics Original Research Article

Scheinerman and Wilf (Amer. Math. Monthly 101 (1994) 939) assert that “an important open problem in the study of graph embeddings is to determine the rectilinear crossing number of the complete graph Kn”. A rectilinear drawing of Kn is an arrangement of n vertices in the plane, every pair of which is connected by an edge that is a line segment. We assume that no three vertices are collinear, and that no three edges intersect in a point unless that point is an endpoint of all three. The rectilinear crossing number of Kn is the fewest number of edge crossings attainable over all rectilinear drawings of Kn. For each n we construct a rectilinear drawing of Kn that has the fewest number of edge crossings and the best asymptotics known to date. Moreover, we give some alternative infinite families of drawings of Kn with good asymptotics. Finally, we mention some old and new open problems.