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We give a new lower bound for the rectilinear crossing number cr(n) of the complete geometric graph K n. We prove that cr(n) ≥

We give a new upper bound for the rectilinear crossing number cr(n) of the complete geometric graph K n. We prove that cr(n) 0.380559 n 4 + Θ(n 3) by means of a new construction based on an iterative duplication strategy starting with a set having a certain structure of halving lines.

Around 1958, Hill conjectured that the crossing number CRg(K<sub>n</sub>) of the complete graph KK<sub>n</sub> is Z(n):=1/4 ⌊ n/2 ⌋ ⌊(n-1)/2⌋ ⌊ (n-2)/2 ⌋ ⌊ (n-3)/2 ⌋ and provided drawings of K<sub>n</sub> with exactly Z(n) crossings. Towards the end of the century, substantially different drawings of… (More)

Let G(v, e) be the set of all simple graphs with v vertices and e edges and let P 2 (G) = d 2 i denote the sum of the squares of the degrees, d 1 ,. .. , d v , of the vertices of G. It is known that the maximum value of P 2 (G) for G ∈ G(v, e) occurs at one or both of two special graphs in G(v, e)—the quasi-star graph or the quasi-complete graph. For each… (More)

The Harary-Hill Conjecture states that the number of crossings in any drawing of the complete graph K n in the plane is at least Z(n) := 1 4 n 2 n−1 2 n−2 2 n−3 2. In this paper, we settle the Harary-Hill conjecture for shellable drawings. We say that a drawing D of K n is s-shellable if there exist a subset S = {v 1 , v 2 ,. .. , v s } of the vertices and… (More)

Recently, Aichholzer, García, Orden, and Ramos derived a remarkably improved lower bound for the number of (≤ k)-edges in an n-point set, and as an immediate corollary an improved lower bound on the rectilinear crossing number of K n. We use simple allowable sequences to extend all their results to the more general setting of simple generalized… (More)

The calculation of the rectilinear crossing number of complete graphs is an important open problem in combinatorial geometry, with important and fruitful connections to other classical problems. Our aim in this work is to survey the body of knowledge around this parameter.

Given a class C of geometric objects and a point set P , a C-matching of P is a set M = {C 1 ,. .. , C k } ⊆ C of elements of C such that each C i contains exactly two elements of P and each element of P lies in at most one C i. If all of the elements of P belong to some C i , M is called a perfect matching. If, in addition, all of the elements of M are… (More)

A generalized configuration is a set of n points and n 2 pseudolines such that each pseudoline passes through exactly two points, two pseudolines intersect exactly once, and no three pseudolines are concurrent. Following the approach of allowable sequences we prove a recursive inequality for the number of (≤ k)-sets for generalized configurations. As a… (More)