Applications of the crossing number
SCG '94 Proceedings of the tenth annual symposium on Computational geometry
Approximation schemes for covering and packing problems in image processing and VLSI
Journal of the ACM (JACM)
On the NP-completeness of the k-colorability problem for triangle-free graphs
Discrete Mathematics
Separators for sphere-packings and nearest neighbor graphs
Journal of the ACM (JACM)
Coloring relatives of intervals on the plane, I: chromatic number versus girth
European Journal of Combinatorics
Label placement by maximum independent set in rectangles
WADS '97 Selected papers presented at the international workshop on Algorithms and data structure
On approximating rectangle tiling and packing
Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms
Improved approximation algorithms for rectangle tiling and packing
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Colouring relatives of intervals on the plane, II: intervals and rays in two directions
European Journal of Combinatorics
Quasi-Planar Graphs Have a Linear Number of Edges
GD '95 Proceedings of the Symposium on Graph Drawing
Graph Drawings with no k Pairwise Crossing Edges
GD '97 Proceedings of the 5th International Symposium on Graph Drawing
Polynomial-time approximation schemes for packing and piercing fat objects
Journal of Algorithms
Geometric Separator Theorems and Applications
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Crossing number, pair-crossing number, and expansion
Journal of Combinatorial Theory Series B
On the maximum number of edges in topological graphs with no four pairwise crossing edges
Proceedings of the twenty-second annual symposium on Computational geometry
Independent set of intersection graphs of convex objects in 2D
Computational Geometry: Theory and Applications
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Intersection graphs of rectangles and segments
General Theory of Information Transfer and Combinatorics
A Separator Theorem for String Graphs and Its Applications
WALCOM '09 Proceedings of the 3rd International Workshop on Algorithms and Computation
On grids in topological graphs
Proceedings of the twenty-fifth annual symposium on Computational geometry
Tangencies between families of disjoint regions in the plane
Proceedings of the twenty-sixth annual symposium on Computational geometry
A separator theorem for string graphs and its applications
Combinatorics, Probability and Computing
Tangencies between families of disjoint regions in the plane
Computational Geometry: Theory and Applications
Maximum independent set in 2-direction outersegment graphs
WG'11 Proceedings of the 37th international conference on Graph-Theoretic Concepts in Computer Science
GD'11 Proceedings of the 19th international conference on Graph Drawing
Coloring planar homothets and three-dimensional hypergraphs
LATIN'12 Proceedings of the 10th Latin American international conference on Theoretical Informatics
Disjoint edges in complete topological graphs
Proceedings of the twenty-eighth annual symposium on Computational geometry
String graphs and incomparability graphs
Proceedings of the twenty-eighth annual symposium on Computational geometry
h-quasi planar drawings of bounded treewidth graphs in linear area
WG'12 Proceedings of the 38th international conference on Graph-Theoretic Concepts in Computer Science
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The intersection graph of a collection C of sets is a graph on the vertex set C, in which C1,C2 ∈ C are joined by an edge if and only if C1 ∩ C2 ≠ Ø. Erdös conjectured that the chromatic number of triangle-free intersection graphs of n segments in the plane is bounded from above by a constant. Here we show that it is bounded by a polylogarithmic function of n, which is the first nontrivial bound for this problem. More generally, we prove that for any t and k, the chromatic number of every Kk-free intersection graph of n curves in the plane, every pair of which have at most t points in common, is at most (ct log n/log k)c log k, where c is an absolute constant and ct only depends on t. We establish analogous results for intersection graphs of convex sets, x-monotone curves, semialgebraic sets of constant description complexity, and sets that can be obtained as the union of a bounded number of sets homeomorphic to a disk. Using a mix of results on partially ordered sets and planar separators, for large k we improve the best known upper bound on the number of edges of a k-quasi-planar topological graph with n vertices, that is, a graph drawn in the plane with curvilinear edges, no k of which are pairwise crossing. As another application, we show that for every ε 0 and for every positive integer t, there exist δ 0 and a positive integer n0 such that every topological graph with n ≥ n0 vertices, at least n1+ε edges, and no pair of edges intersecting in more than t points, has at least nδ pairwise intersecting edges.