Combinatorial complexity bounds for arrangements of curves and spheres
Discrete & Computational Geometry - Special issue on the complexity of arrangements
Which crossing number is it anyway?
Journal of Combinatorial Theory Series B
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
Computing crossing number in linear time
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
On the number of tetrahedra with minimum, unit, and distinct volumes in three-space
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
On the decay of crossing numbers
Journal of Combinatorial Theory Series B
On the decay of crossing numbers
GD'06 Proceedings of the 14th international conference on Graph drawing
Improvement on the decay of crossing numbers
GD'07 Proceedings of the 15th international conference on Graph drawing
Acyclic orientation of drawings
SWAT'06 Proceedings of the 10th Scandinavian conference on Algorithm Theory
Crossing stars in topological graphs
JCDCG'04 Proceedings of the 2004 Japanese conference on Discrete and Computational Geometry
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Twenty years ago, Ajtai, Chvátal, Newborn, Szemerédi, and, independently, Leighton discovered that the crossing number of any graph with v vertices and e4v edgesis at least ce3/v2, where c0 is an absolute constant. This result, known as the 'Crossing Lemma,' has found many important applications in discrete and computational geometry. It is tightup to a multiplicative constant. Here we improve the best known value of the constant by showing that the result holds with c1024/318270.032. The proof has two new ingredients, interesting on their own right. We show that (1) if a graph can be drawn in the plane so that every edge crosses at most 3 others, then its number of edges cannot exceed 5.5(v-2); and (2) the crossing number of any graph is at least 73e - 253(v-2). Both bounds are tight up to anadditive constant (the latter one in the range 4v ≤ e ≤ 5v).