Results 1  10
of
16
A lower bound for the rectilinear crossing number
 Graphs and Combinatorics
, 2003
"... We give a new lower bound for the rectilinear crossing number cr(n) of the complete geometric graph Kn. Weprovethatcr(n) ≥ 1 ¥ ¦ ¥ ¦ ¥ ¦ ¥ ¦ n n−1 n−2 n−3 andweextendtheproofof ..."
Abstract

Cited by 32 (14 self)
 Add to MetaCart
We give a new lower bound for the rectilinear crossing number cr(n) of the complete geometric graph Kn. Weprovethatcr(n) ≥ 1 ¥ ¦ ¥ ¦ ¥ ¦ ¥ ¦ n n−1 n−2 n−3 andweextendtheproofof
Convex Quadrilaterals and kSets
 Contemporary Mathematics Series, 342, AMS 2004
, 2003
"... Introduction Let S be a set of n points in general position in the plane, i.e. no three points are collinear. Four points in S may or may not form the vertices of a convex quadrilateral; if they do, we call this subset of 4 elements convex. We are interested in the number of convex 4element subset ..."
Abstract

Cited by 18 (0 self)
 Add to MetaCart
Introduction Let S be a set of n points in general position in the plane, i.e. no three points are collinear. Four points in S may or may not form the vertices of a convex quadrilateral; if they do, we call this subset of 4 elements convex. We are interested in the number of convex 4element subsets. This can of course be as large as , if S is in convex position, but what is its minimum? Another way of stating the problem is to find the rectilinear crossing number of the complete ngraph K n , i.e., to determine the minimum number of crossings in a drawing of K n in the plane with straight edges and the nodes in general position. We note here that the rectilinear crossing number of complete graphs also determines the rectilinear crossing number of random graphs (provided the probability for an edge to appear is at least ln n ), as was shown by Spencer and Toth [13]. It is easy to see that for n = 5 we get at least one convex 4element subset, from which it follows by straigh
New lower bounds for the number of (≤ k)edges and the rectilinear crossing number of Kn, Discrete Comput
 Geom
"... We provide a new lower bound on the number of ( ≤ k)edges of a set of n points in the plane in general position. We show that for 0 ≤ k ≤ ⌊ n−2 2 ⌋ the number of ( ≤ k)edges is at least ( ) k ∑ k + 2 Ek(S) ≥ 3 + (3j − n + 3), 2 j= ⌊ n 3 ⌋ which, for k ≥ ⌊ n 3 ⌋, improves the previous best lower ..."
Abstract

Cited by 18 (3 self)
 Add to MetaCart
We provide a new lower bound on the number of ( ≤ k)edges of a set of n points in the plane in general position. We show that for 0 ≤ k ≤ ⌊ n−2 2 ⌋ the number of ( ≤ k)edges is at least ( ) k ∑ k + 2 Ek(S) ≥ 3 + (3j − n + 3), 2 j= ⌊ n 3 ⌋ which, for k ≥ ⌊ n 3 ⌋, improves the previous best lower bound in [7]. As a main consequence, we obtain a new lower bound on the rectilinear crossing number of the complete graph or, in other words, on the minimum number of convex quadrilaterals determined by n points in the plane in general position. We show that the crossing number is at least
Geometric drawings of Kn with few crossings
 J. Combin. Theory Ser. A
"... We give a new upper bound for the rectilinear crossing number cr(n) of the complete geometric graph Kn. Weprovethatcr(n) ≤ 0.380559 ¡ ¢ n 3 + Θ(n) by means of a new construction 4 based on an iterative duplication strategy starting with a set having a certain structure of halving lines. ..."
Abstract

Cited by 14 (11 self)
 Add to MetaCart
We give a new upper bound for the rectilinear crossing number cr(n) of the complete geometric graph Kn. Weprovethatcr(n) ≤ 0.380559 ¡ ¢ n 3 + Θ(n) by means of a new construction 4 based on an iterative duplication strategy starting with a set having a certain structure of halving lines.
Planar decompositions and the crossing number of graphs with an excluded minor
 IN GRAPH DRAWING 2006; LECTURE NOTES IN COMPUTER SCIENCE 4372
, 2007
"... Tree decompositions of graphs are of fundamental importance in structural and algorithmic graph theory. Planar decompositions generalise tree decompositions by allowing an arbitrary planar graph to index the decomposition. We prove that every graph that excludes a fixed graph as a minor has a planar ..."
Abstract

Cited by 14 (1 self)
 Add to MetaCart
Tree decompositions of graphs are of fundamental importance in structural and algorithmic graph theory. Planar decompositions generalise tree decompositions by allowing an arbitrary planar graph to index the decomposition. We prove that every graph that excludes a fixed graph as a minor has a planar decomposition with bounded width and a linear number of bags. The crossing number of a graph is the minimum number of crossings in a drawing of the graph in the plane. We prove that planar decompositions are intimately related to the crossing number. In particular, a graph with bounded degree has linear crossing number if and only if it has a planar decomposition with bounded width and linear order. It follows from the above result about planar decompositions that every graph with bounded degree and an excluded minor has linear crossing number. Analogous results are proved for the convex and rectilinear crossing numbers. In particular, every graph with bounded degree and bounded treewidth has linear convex crossing number, and every K3,3minorfree graph with bounded degree has linear rectilinear crossing number.
The Rectilinear Crossing Number of K_10 is 62
 Electron. J. Combin., 8(1):Research Paper
, 2000
"... The rectilinear crossing number of a graph G is the minimum number of edge crossings that can occur in any drawing of G in which the edges are straight line segments and no three vertices are collinear. This number has been known for G = K n if n # 9. Using a combinatorial argument we show that fo ..."
Abstract

Cited by 11 (0 self)
 Add to MetaCart
The rectilinear crossing number of a graph G is the minimum number of edge crossings that can occur in any drawing of G in which the edges are straight line segments and no three vertices are collinear. This number has been known for G = K n if n # 9. Using a combinatorial argument we show that for n =10the number is 62.
On ksets, convex quadrilaterals, and the rectilinear crossing number of K_n
 of Kn, Discr. Comput. Geom
, 2004
"... We use circular sequences to give an improved lower bound on the minimum number of (# k) sets in a set of points in general position. We then use this to show that if S is a set of n points in general position, then the number #(S) of convex quadrilaterals determined by the points in S is at leas ..."
Abstract

Cited by 11 (9 self)
 Add to MetaCart
We use circular sequences to give an improved lower bound on the minimum number of (# k) sets in a set of points in general position. We then use this to show that if S is a set of n points in general position, then the number #(S) of convex quadrilaterals determined by the points in S is at least 0.37533 ). This in turn implies that the rectilinear crossing number cr(Kn ) of the complete graph Kn is at least 0.37533 ), and that Sylvester's Four Point Problem Constant is at least 0.37533. These improved bounds refine results recently obtained by Abrego and FernandezMerchant, and by Lovasz, Vesztergombi, Wagner and Welzl.
On the Rectilinear Crossing Number of Complete Graphs
 Proc. 14th ACMSIAM Sympos. Discr. Alg
, 2003
"... We prove a lower bound of 0:3288 for the rectilinear crossing number cr(Kn ) of a complete graph on n vertices, or in other words, for the minimum number of convex quadrilaterals in any set of n points in general position in the Euclidean plane. As we see it, the main contribution of this pa ..."
Abstract

Cited by 8 (1 self)
 Add to MetaCart
We prove a lower bound of 0:3288 for the rectilinear crossing number cr(Kn ) of a complete graph on n vertices, or in other words, for the minimum number of convex quadrilaterals in any set of n points in general position in the Euclidean plane. As we see it, the main contribution of this paper is not so much the concrete numerical improvement over earlier bounds, as the novel method of proof, which is not based on bounding cr(Kn ) for some small n.
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 ..."
Abstract

Cited by 6 (5 self)
 Add to MetaCart
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.