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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.

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) ≥

Around 1958, Hill described how to draw the complete graph K n with Z (n) := 1 4 𢶀 n 2 𢶀 n − 1 2 𢶀 𢶀 n − 2 2 𢶀 𢶀 n − 3 2 crossings, and conjectured that the crossing number cr(K n) of K n is exactly Z(n). This is also known as Guy's conjecture as he later popularized it. Towards the end of the century, substantially different drawings of K n with Z(n)… (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)

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)

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)

Even the most supercial glance at the vast majority of crossing-minimal geometric drawings of K n reveals two hard-to-miss features. First, all such drawings appear to be 3-fold symmetric (or simply 3-symmetric). And second, they all are 3-decomposable, that is, there is a triangle T enclosing the drawing, and a balanced partition A, B, C of the underlying… (More)

Let P be a set of points in general position in the plane. Join all pairs of points in P with straight line segments. The number of segment-crossings in such a drawing, denoted by cr(P), is the rectilinear crossing number of P. A halving line of P is a line passing though two points of P that divides the rest of the points of P in (almost) half. The number… (More)