Disjoint edges in topological graphs

Authors:
János Pach;Géza Tóth
Affiliations:
City College, CUNY and Courant Institute of Mathematical Sciences, New York University, New York, NY;Rényi Institute of the Hungarian Academy of Sciences, Budapest, Hungary
Venue:
IJCCGGT'03 Proceedings of the 2003 Indonesia-Japan joint conference on Combinatorial Geometry and Graph Theory
Year:
2003

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Abstract

A topological graph G is a graph drawn in the plane so that its edges are represented by Jordan arcs. G is called simple, if any two edges have at most one point in common. It is shown that the maximum number of edges of a simple topological graph with n vertices and no k pairwise disjoint edges is O(nlog4k−8n) edges. The assumption that G is simple cannot be dropped: for every n, there exists a complete topological graph of n vertices, whose any two edges cross at most twice.