Discrete Applied Mathematics - Combinatorics and complexity
Which Crossing Number is it, Anyway?
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Coloring Kk-free intersection graphs of geometric objects in the plane
European Journal of Combinatorics
Disjoint edges in topological graphs
IJCCGGT'03 Proceedings of the 2003 Indonesia-Japan joint conference on Combinatorial Geometry and Graph Theory
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A topological graph is a graph drawn in the plane so that its vertices are represented by points, and its edges are represented by Jordan curves connecting the corresponding points, with the property that any two curves have at most one point in common. We define two canonical classes of topological complete graphs, and prove that every topological complete graph with n vertices has a canonical subgraph of size at least c log log n, which belongs to one of these classes. We also show that every complete topological graph with n vertices has a noncrossing subgraph isomorphic to any fixed tree with at most c log1/6 n vertices.