Degenerate crossing numbers

Authors:
János Pach;Géza Tóth
Affiliations:
City College, CUNY and Rényi Institute;Rényi Institute, Hungarian Academy of Sciences
Venue:
Proceedings of the twenty-second annual symposium on Computational geometry
Year:
2006

Citing 2
Cited 1

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Abstract

Let G be a graph with n vertices and e ≥ 4n edges, drawn in the plane in such a way that if two or more edges (arcs) share an interior point p ,then they must properly cross one another at p. It is shown that the number of crossing points, counted without multiplicity, is at least constant times e and that the order of magnitude of this bound cannot be improved. If, in addition, two edges are allowed to cross only at most once, then the number of crossing points must exceed constant times (e/n)4.