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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 ..."
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Cited by 25 (2 self)
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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
On Distinct Distances from a Vertex of a Convex Polygon
"... Given a set P of n points in convex position in the plane, we prove that there exists a point p 2 P such that the number of distinct distances from p is at least d(13n  6)/36e. The best previous bound, dn=3e, from 1952, is due to Leo Moser. ..."
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Cited by 4 (1 self)
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Given a set P of n points in convex position in the plane, we prove that there exists a point p 2 P such that the number of distinct distances from p is at least d(13n  6)/36e. The best previous bound, dn=3e, from 1952, is due to Leo Moser.
Distinct triangle areas in a planar point set
, 2007
"... Abstract Erd""os, Purdy, and Straus conjectured that the number of distinct (nonzero) areas of the trianglesdetermined by n noncollinear points in the plane is at least b n12 c, which is attained for dn/2e andrespectively b n/2c equally spaced points lying on two parallel lines. We show t ..."
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Cited by 3 (0 self)
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Abstract Erd""os, Purdy, and Straus conjectured that the number of distinct (nonzero) areas of the trianglesdetermined by n noncollinear points in the plane is at least b n12 c, which is attained for dn/2e andrespectively b n/2c equally spaced points lying on two parallel lines. We show that this number isat least 17 38 n O(1) ss 0.4473n. The best previous bound, (p2 1)n O(1) ss 0.4142n, whichdates back to 1982, follows from the combination of a result of Burton and Purdy [5] and Ungar's theorem [23] on the number of distinct directions determined by n noncollinear points in the plane. 1 Introduction Let S be a finite set of points in the plane. Consider the (nondegenerate) triangles determined by triplesof points of S. There are at most \Gamma n3\Delta triangles, some of which may have the same area. Denote by g(S)the number of distinct (nonzero) areas of the triangles determined by
Improving the Crossing Lemma by finding more crossings
 in sparse graphs, 20th ACM Symposium on Computational Geometry, ACM
, 2004
"... in sparse graphs ..."
A Bipartite Strengthening of the Crossing Lemma
"... Abstract. The celebrated Crossing Lemma states that, in every drawing of a graph with n vertices and m ≥ 4n edges there are at least Ω(m 3 /n 2) pairs of crossing edges; or equivalently, there is an edge that crosses Ω(m 2 /n 2) other edges. We strengthen the Crossing Lemma for drawings in which any ..."
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Cited by 1 (1 self)
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Abstract. The celebrated Crossing Lemma states that, in every drawing of a graph with n vertices and m ≥ 4n edges there are at least Ω(m 3 /n 2) pairs of crossing edges; or equivalently, there is an edge that crosses Ω(m 2 /n 2) other edges. We strengthen the Crossing Lemma for drawings in which any two edges cross in at most O(1) points. We prove for every k ∈ N that every graph G with n vertices and m ≥ 3n edges drawn in the plane such that any two edges intersect in at most k points has two disjoint subsets of edges, E1 and E2, each of size at least ckm 2 /n 2, such that every edge in E1 crosses all edges in E2, where ck> 0 only depends on k. This bound is best possible up to the constant ck for every k ∈ N. We also prove that every graph G with n vertices and m ≥ 3n edges drawn in the plane with xmonotone edges has disjoint subsets of edges, E1 and E2, each of size Ω(m 2 /(n 2 polylog n)), such that every edge in E1 crosses all edges in E2. On the other hand, we construct xmonotone drawings of bipartite dense graphs where the largest such subsets E1 and E2 have size O(m 2 /(n 2 log(m/n))). 1
On
"... distinct distances among points in general position and other related problems ..."
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distinct distances among points in general position and other related problems