On the decay of crossing numbers

The crossing number cr(G) of a graph G is the minimum number of crossings over all drawings of G in the plane. In 1993, Richter and Thomassen [B. Richter, C. Thomassen, Minimal graphs with crossing number at least k, J. Combin. Theory Ser. B 58 (1993) 217–224] conjectured that there is a constant c such that every graph G with crossing number k has an edge e such that . They showed only that G always has an edge e with . We prove that for every fixed ϵ>0, there is a constant n0 depending on ϵ such that if G is a graph with n>n0 vertices and m>n1+ϵ edges, then G has a subgraph G′ with at most edges such that .