Results 1  10
of
52
Crossing numbers and hard Erdős problems in discrete geometry
 COMBINATORICS, PROBABILITY AND COMPUTING
, 1997
"... We show that an old but not wellknown lower bound for the crossing number of a graph yields short proofs for a number of bounds in discrete plane geometry which were considered hard before: the number of incidences among points and lines, the maximum number of unit distances among n points, the min ..."
Abstract

Cited by 113 (1 self)
 Add to MetaCart
We show that an old but not wellknown lower bound for the crossing number of a graph yields short proofs for a number of bounds in discrete plane geometry which were considered hard before: the number of incidences among points and lines, the maximum number of unit distances among n points, the minimum number of distinct distances among n points.
Proofs from the Book
, 1998
"... Paul Erdős liked to talk about The Book, in which God maintains the perfect proofs for mathematical theorems, following the dictum of G. H. Hardy that there is no permanent place for ugly mathematics. Erdős also said that you need not believe in God but, as a mathematician, you should believe in The ..."
Abstract

Cited by 57 (1 self)
 Add to MetaCart
Paul Erdős liked to talk about The Book, in which God maintains the perfect proofs for mathematical theorems, following the dictum of G. H. Hardy that there is no permanent place for ugly mathematics. Erdős also said that you need not believe in God but, as a mathematician, you should believe in The Book. A few years ago, we suggested to him to write up a first (and very modest) approximation to The Book. He was enthusiastic about the idea and, characteristically, went to work immediately, filling page after page with his suggestions. Our book was supposed to appear in March 1998 as a present to Erdős ’ 85th birthday. With Paul’s unfortunate death in the summer of 1997, he is not listed as a coauthor. Instead this book is dedicated to his memory. We have no definition or characterization of what constitutes a proof from The Book: all we offer here is the examples that we have selected, hoping that our readers will share our enthusiasm about brilliant ideas, clever insights and wonderful observations. We also hope that our readers will
Applications of the crossing number
 In Proc. 10th Annu. ACM Sympos. Comput. Geom
, 1994
"... Abstract. The crossing number of a graph G is the minimum number of crossings in a drawing of G. The determination of the crossing number is an NPcomplete problem. We present two general lower bounds for the crossing number, and survey their applications and generalizations. 1 ..."
Abstract

Cited by 28 (6 self)
 Add to MetaCart
Abstract. The crossing number of a graph G is the minimum number of crossings in a drawing of G. The determination of the crossing number is an NPcomplete problem. We present two general lower bounds for the crossing number, and survey their applications and generalizations. 1
Improving the Crossing Lemma by Finding More Crossings in Sparse Graphs
, 2006
"... Twenty years ago, Ajtai et al. and, independently, Leighton discovered that the crossing number of any graph with v vertices and e> 4v edges is at least ce3 /v2, where c> 0 is an absolute constant. This result, known as the “Crossing Lemma, ” has found many important applications in discrete and com ..."
Abstract

Cited by 25 (2 self)
 Add to MetaCart
Twenty years ago, Ajtai et al. and, independently, Leighton discovered that the crossing number of any graph with v vertices and e> 4v edges is at least ce3 /v2, where c> 0 is an absolute constant. This result, known as the “Crossing Lemma, ” has found many important applications in discrete and computational geometry. It is tight up to a multiplicative constant. Here we improve the best known value of the constant by showing that the result holds with c> 1024/31827> 0.032. The proof has two new ingredients, interesting in their own right. We show that (1) if a graph can be drawn in the plane so that every edge crosses at most three others, then its number of edges cannot exceed 5.5(v − 2); and (2) the crossing number of any graph is at least 7 25 e − (v − 2). Both bounds are tight upt o
Toward the Rectilinear Crossing Number of K_n : New Embeddings, Upper Bounds, and Asymptotics
, 2000
"... Scheinerman and Wilf [SW94] assert that "an important open problem in the study of graph embeddings is to determine the rectilinear crossing number of the complete graph Kn ." A rectilinear embedding or drawing of Kn is an arrangement of n vertices in the plane, every pair of which is connected by ..."
Abstract

Cited by 22 (1 self)
 Add to MetaCart
Scheinerman and Wilf [SW94] assert that "an important open problem in the study of graph embeddings is to determine the rectilinear crossing number of the complete graph Kn ." A rectilinear embedding or drawing of Kn is an arrangement of n vertices in the plane, every pair of which is connected by an edge that is a line segment. We assume that no three vertices are collinear. The rectilinear crossing number of Kn is the fewest number of edge crossings attainable over all planar rectilinear embeddings of Kn . For each n we construct a rectilinear embedding of Kn that has the fewest number of edge crossings and the best asymptotics known to date. Moreover, we give some alternative infinite families of embeddings of Kn with good asymptotics. Finally, we mention some old and new open problems.
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
On VLSI Layouts Of The Star Graph And Related Networks
, 1994
"... . We prove that the minimal VLSI layout of the arrangement graph A(n; k) occupies \Theta(n!=(n \Gamma k \Gamma 1)!) 2 area. As a special case we obtain an optimal layout for the star graph S n with the area \Theta(n!) 2 : This answers an open problem posed by Akers, Harel and Krishnamurthy [1]. ..."
Abstract

Cited by 18 (3 self)
 Add to MetaCart
. We prove that the minimal VLSI layout of the arrangement graph A(n; k) occupies \Theta(n!=(n \Gamma k \Gamma 1)!) 2 area. As a special case we obtain an optimal layout for the star graph S n with the area \Theta(n!) 2 : This answers an open problem posed by Akers, Harel and Krishnamurthy [1]. The method is also applied to the pancake graph. The results provide optimal upper and lower bounds for crossing numbers of the above graphs. Key Words: area, arrangement graph, congestion, crossing number, embedding, layout, pancake graph, star graph, VLSI 1
Thirteen Problems on Crossing Numbers
, 2000
"... The crossing number of a graph G is the minimum number of crossings in a drawing of G. We introduce several variants of this definition, and present a list of related open problems. The first item is Zarankiewicz's classical conjecture about crossing numbers of complete bipartite graphs, the la ..."
Abstract

Cited by 15 (0 self)
 Add to MetaCart
The crossing number of a graph G is the minimum number of crossings in a drawing of G. We introduce several variants of this definition, and present a list of related open problems. The first item is Zarankiewicz's classical conjecture about crossing numbers of complete bipartite graphs, the last ones are new and less carefully tested. In Section 5, we state some conjectures about the expected values of various crossing numbers of random graphs, and prove a large deviation result.
On Crossing Numbers Of Hypercubes And Cube Connected Cycles
, 1993
"... . We prove tight bounds for crossing numbers of hypercube and cube connected cycles (CCC) graphs. Key Words: crossing number, cube connected cycles, hypercube, lower bound CR Categories: F.1.2 [Modes of Computations] Parallelism, G.2.2. [Graph Theory] Network problems 1 ..."
Abstract

Cited by 15 (3 self)
 Add to MetaCart
. We prove tight bounds for crossing numbers of hypercube and cube connected cycles (CCC) graphs. Key Words: crossing number, cube connected cycles, hypercube, lower bound CR Categories: F.1.2 [Modes of Computations] Parallelism, G.2.2. [Graph Theory] Network problems 1