Topological graphs with no self-intersecting cycle of length 4

Authors:
Rom Pinchasi;Radoa Radoicic
Affiliations:
Massachusetts Institute of Technology, Cambridge, MA;Massachusetts Institute of Technology, Cambridge, MA
Venue:
Proceedings of the nineteenth annual symposium on Computational geometry
Year:
2003

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Abstract

Let G be a topological graph on n vertices in the plane, i.e., a graph drawn in the plane with its vertices represented as points and its edges represented as Jordan arcs connecting pairs of points. It is shown that if no two edges of any cycle of length 4 in G cross an odd number of times, then |E(G)|=O(n8/5).