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 cr ( G − e ) ⩾ k − c k . They showed only that G always has an edge e with cr ( G − e ) ⩾ 2 5 cr ( G ) − O ( 1 ) . We prove that for every fixed ϵ > 0 , there is a constant n 0 depending on ϵ such that if G is a graph with n > n 0 vertices and m > n 1 + ϵ edges, then G has a subgraph G ′ with at most ( 1 − ϵ 24 ) m edges such that cr ( G ′ ) ⩾ ( 1 28 − o ( 1 ) ) cr ( G ) .