Degenerate crossing numbers

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 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. © 2009 Springer Science+Business Media, LLC.