Some provably hard crossing number problems
Discrete & Computational Geometry
New results on rectilinear crossing numbers and plane embeddings
Journal of Graph Theory
Bounds for rectilinear crossing numbers
Journal of Graph Theory
New bounds on crossing numbers
SCG '99 Proceedings of the fifteenth annual symposium on Computational geometry
On the rectilinear crossing number of complete graphs
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Note: Geometric drawings of Kn with few crossings
Journal of Combinatorial Theory Series A
3-symmetric and 3-decomposable geometric drawings of Kn
Discrete Applied Mathematics
GD'04 Proceedings of the 12th international conference on Graph Drawing
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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 gave some alternative infinite families of drawings of Kn, with good asymptotics. Finally, we mention some old and new open problems.