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An extended lower bound on the number of ( ≤ k)edges to generalized configurations of points and the pseudolinear crossing number of K_n
, 2007
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The maximum number of halving lines and the rectilinear crossing number of Kn for n ≤ 27
, 2007
"... For n ≤ 27 we present exact values for the maximum number h(n) of halving lines and eh(n) of halving pseudolines, determined by n points in the plane. For this range of values of n we also present exact values of the rectilinear cr(n) and the pseudolinear ecr(n) crossing number of the complete ..."
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Cited by 7 (5 self)
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For n ≤ 27 we present exact values for the maximum number h(n) of halving lines and eh(n) of halving pseudolines, determined by n points in the plane. For this range of values of n we also present exact values of the rectilinear cr(n) and the pseudolinear ecr(n) crossing number of the complete
On (≤ k)pseudoedges in generalized configurations and the pseudolinear crossing number of K_n
, 2006
"... It is known that every generalized configuration with n points has at least 3 ` k+2 2 ´ ( ≤ k)–pseudoedges, and that this bound is tight for k ≤ n/3 − 1. Here we show that this bound is no longer tight for (any) k> n/3 − 1. As a corollary, we prove that the usual and the pseudolinear (and hence the ..."
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Cited by 6 (5 self)
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It is known that every generalized configuration with n points has at least 3 ` k+2 2 ´ ( ≤ k)–pseudoedges, and that this bound is tight for k ≤ n/3 − 1. Here we show that this bound is no longer tight for (any) k> n/3 − 1. As a corollary, we prove that the usual and the pseudolinear (and hence the rectilinear) crossing numbers of the complete graph Kn are different for every n ≥ 10. It has been noted that all known optimal rectilinear drawings of Kn share a triangular–like property, which we abstract into the concept of 3–decomposability. We give a lower bound for the crossing numbers of all pseudolinear drawings of Kn that satisfy this property. This bound coincides with the best general lower bound known for the rectilinear crossing number of Kn, established recently in a groundbreaking work by Aichholzer, García, Orden, and Ramos. We finally use these results to calculate the pseudolinear (which happen to coincide with the rectilinear) crossing numbers of Kn for n ≤ 12 and n = 15.
New results on lower bounds for the number of (≤ k)facets
, 2008
"... In this paper we present three different results dealing with the number of ( ≤ k)facets of a set of points: 1. We give structural properties of sets in the plane that achieve the optimal lower bound 3 () k+2 2 of ( ≤ k)edges for a fixed 0 ≤ k ≤ ⌊n/3 ⌋ − 1; 2. We give a simple construction showing ..."
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Cited by 3 (0 self)
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In this paper we present three different results dealing with the number of ( ≤ k)facets of a set of points: 1. We give structural properties of sets in the plane that achieve the optimal lower bound 3 () k+2 2 of ( ≤ k)edges for a fixed 0 ≤ k ≤ ⌊n/3 ⌋ − 1; 2. We give a simple construction showing that the lower bound 3 ( ) ( n k+2 k−⌊
THE RECTILINEAR CROSSING NUMBER OF Kn: CLOSING IN (OR ARE WE?)
"... Abstract. 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. 1. ..."
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Cited by 2 (0 self)
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Abstract. 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. 1.
3symmetric and 3decomposable geometric drawings of Kn
, 2009
"... Even the most super cial glance at the vast majority of crossingminimal geometric drawings of Kn reveals two hardtomiss features. First, all such drawings appear to be 3fold symmetric (or simply 3symmetric). And second, they all are 3decomposable, that is, there is a triangle T enclosing the d ..."
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Cited by 1 (1 self)
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Even the most super cial glance at the vast majority of crossingminimal geometric drawings of Kn reveals two hardtomiss features. First, all such drawings appear to be 3fold symmetric (or simply 3symmetric). And second, they all are 3decomposable, that is, there is a triangle T enclosing the drawing, and a balanced partition A, B, C of the underlying set of points P, such that the orthogonal projections of P onto the sides of T show A between B and C on one side, B between A and C on another side, and C between A and B on the third side. In fact, we conjecture that all optimal drawings are 3decomposable, and that there are 3symmetric optimal constructions for all n multiple of 3. In this paper, we show that any 3decomposable geometric drawing of Kn has at least 0.380029 () n 3 4 + Θ(n) crossings. On the other hand, we produce 3symmetric and 3decomposable drawings that improve the general upper bound for the rectilinear crossing number of Kn to 0.380488 () n 3 4 +Θ(n). We also give explicit 3symmetric and 3decomposable constructions for n < 100 that are at least as good as those previously known. 1
Recent developments on the number of ( ≤ k)sets, halving lines, and the rectilinear crossing number of Kn.
"... We present the latest developments on the number of ( ≤ k)sets and halving lines for (generalized) configurations of points; as well as the rectilinear and pseudolinear crossing numbers of Kn. In particular, we define perfect generalized configurations on n points as those whose number of ( ≤ k)se ..."
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Cited by 1 (0 self)
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We present the latest developments on the number of ( ≤ k)sets and halving lines for (generalized) configurations of points; as well as the rectilinear and pseudolinear crossing numbers of Kn. In particular, we define perfect generalized configurations on n points as those whose number of ( ≤ k)sets is exactly 3 ¡ ¢ k+1 for all k ≤ n/3. We conjecture that for each n there 2 is a perfect configuration attaining the maximum number of ( ≤ k)sets and the pseudolinear crossing number of Kn. We prove that for any k ≤ n/2 thenumberof(≤k)sets is at least 3 ¡ ¢ ¡ ¢ ¡ ¢ k+1 k−bn/3c+1 k−d4n/9e+1 +3 +18 − O (n). This in turn implies that the pseudolinear (and 2 2
unknown title
, 2006
"... An extended lower bound on the number of ( ≤ k)–edges to generalized configurations of points and the pseudolinear crossing number of Kn ..."
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An extended lower bound on the number of ( ≤ k)–edges to generalized configurations of points and the pseudolinear crossing number of Kn