On the Decay of Crossing Numbers

The crossing number cr(G) of a graphG is the minimum number of crossings over all drawings of G in the plane. In 1993, Richter and Thomassen [RT93] 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 ǫ > 0, there is a constant n0 depending on ǫ such that if G is a graph with n > n0 vertices and m > n 1+ǫ edges, then G has a subgraph G with at most (1− 1 24ǫ )m edges such that cr(G) ≥ ( 1 28 − o(1))cr(G).