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- Bernardo M. Ábrego, Silvia Fernández-Merchant
- Graphs and Combinatorics
- 2005

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) ≥

- Bernardo M. Ábrego, Silvia Fernández-Merchant
- J. Comb. Theory, Ser. A
- 2007

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.

- Bernardo M. Ábrego, Oswin Aichholzer, Silvia Fernández-Merchant, Pedro Ramos, Gelasio Salazar
- Discrete & Computational Geometry
- 2012

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)

- Bernardo M. Ábrego, Oswin Aichholzer, Silvia Fernández-Merchant, Pedro Ramos, Gelasio Salazar
- Discrete & Computational Geometry
- 2014

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)

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.

Let Mm,n(0, 1) denote the set of all m × n (0,1)-matrices and let In this paper we exhibit some new formulas for G(m, n) where n ≡ −1 (mod 4). Specifically, for m = nt + r where 0 ≤ r < n, we show that for all sufficiently large t, G(nt + r, n) is a polynomial in t of degree n that depends on the characteristic polynomial of the adjacency matrix of a… (More)

- Bernardo M. Ábrego, József Balogh, Silvia Fernández-Merchant, Jesús Leaños, Gelasio Salazar
- J. Comb. Theory, Ser. A
- 2008

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)

- Bernardo M. Ábrego, Esther M. Arkin, +4 authors Jorge Urrutia
- Discrete & Computational Geometry
- 2009

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)

- Bernardo M. Ábrego, Silvia Fernández-Merchant
- Discrete & Computational Geometry
- 2000