On the Degenerate Crossing Number

The degenerate crossing number\({\text{ cr}^{*}}(G)\) 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 Toth who showed that for a graph \(G\) with \(n\) vertices and \(e \ge 4n\) edges \({\text{ cr}^{*}}(G)=\Omega \big (e^4 / n^4\big )\). In this paper we completely resolve the main open question about degenerate crossing numbers and show that \({\text{ cr}^{*}}(G)=\Omega \big (e^3 / n^2 \big )\), provided that \(e \ge 4n\). 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.