Which crossing number is it anyway?
Journal of Combinatorial Theory Series B
Journal of Combinatorial Theory Series A
Planar Separators and the Euclidean Norm
SIGAL '90 Proceedings of the International Symposium on Algorithms
On the Number of Incidences Between Points and Curves
Combinatorics, Probability and Computing
Crossing Numbers and Hard Erdös Problems in Discrete Geometry
Combinatorics, Probability and Computing
Crossing number, pair-crossing number, and expansion
Journal of Combinatorial Theory Series B
Topological Graphs with No Large Grids
Graphs and Combinatorics
Improving the Crossing Lemma by Finding More Crossings in Sparse Graphs
Discrete & Computational Geometry
String graphs and incomparability graphs
Proceedings of the twenty-eighth annual symposium on Computational geometry
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Let G=(V,E) be a graph with n vertices and m=4n edges drawn in the plane. The celebrated Crossing Lemma states that G has at least @W(m^3/n^2) pairs of crossing edges; or equivalently, there is an edge that crosses @W(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. An @?-grid in the drawing of G is a pair E"1,E"2@?E of disjoint edge subsets each of size @? such that every edge in E"1 intersects every edge in E"2. If every pair of edges of G intersect in at most k points, then G contains an @?-grid with @?=c"km^2/n^2, where c"k0 only depends on k. Without any assumption on the number of points in which edges cross, we prove that G contains an @?-grid with @?=m^2/n^2polylog(m/n). If G is dense, that is, m=@Q(n^2), our proof demonstrates that G contains an @?-grid with @?=@W(n^2/logn). We show that this bound is best possible up to a constant factor by constructing a drawing of the complete bipartite graph K"n","n using expander graphs in which the largest @?-grid satisfies @?=@Q(n^2/logn).