On the Degenerate Crossing Number

The degenerate crossing number cr∗(G) of a graph G is the minimum number of crossing points of edges in any drawing of G as a simple topological graph in the plane. This notion was introduced by Pach and Tóth who showed that for a graph G with n vertices and e ≥ 4n edges cr∗(G) = Ω(e/n). In this paper we completely resolve the main open question about degenerate crossing numbers and show that cr∗(G) = Ω(e/n), provided that e ≥ 4n. This bound is best possible (apart for the multiplicative constant) as it matches the tight lower bound for the standard crossing number of a graph.