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 4 /n 4 ). In this paper we completely resolve the main open question about degenerate crossing numbers and show that cr * (G) = Ω(e 3 /n 2 ), 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.
The corresponding simple degenerate crossing number is Ω(m 3 /n 2 ) for graphs with m ≥ 4n edges REF .
On the degenerate crossing number
Discrete & Computational Geometry
Computer Science Mathematics