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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 Kn. We prove that cr(n) ≥ 14 ¥ n 2 ¦ ¥ n−1 2 ¦ ¥ n−2 2 ¦ ¥ n−3 2 ¦ and we extend the proof of the result to pseudolinear drawings of Kn.

Let G(v, e) be the set of all simple graphs with v vertices and e edges and let P2(G) = ∑ d i denote the sum of the squares of the degrees, d1, . . . , dv, of the vertices of G. It is known that the maximum value of P2(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 pair… (More)

- 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)

- 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 Kn. We prove that cr(n) 0.380559 (n 4 )+Θ(n3) by means of a new construction based on an iterative duplication strategy starting with a set having a certain structure of halving lines. © 2006 Elsevier Inc. All rights reserved.

- Bernardo M. Ábrego, Esther M. Arkin, +4 authors Jorge Urrutia
- JCDCG
- 2004

- 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 = {C1, . . . , Ck} ⊆ C of elements of C such that each Ci contains exactly two elements of P and each element of P lies in at most one Ci. If all of the elements of P belong to some Ci, M is called a perfect matching. If, in addition, all of the elements of M are pairwise… (More)

- Bernardo M. Ábrego, Mario Cetina, Silvia Fernández-Merchant, Jesús Leaños, Gelasio Salazar
- Discrete Applied Mathematics
- 2010

Even the most super cial glance at the vast majority of crossing-minimal geometric drawings of Kn 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)

- 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 Kn 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 Kn is s-shellable if there exist a subset S = {v1, v2, . . . , vs} of the… (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.

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