On the decay of crossing numbers

Abstract

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 [RT93] conjectured that there is a constant c such that every graph G with crossing number k has an edge c such that cr(G e) ≥ k - c√k. They showed only that G always has an edge e with cr(G - e) ≥ 2/5cr(G) - O(1). 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 (1 - 1/24e)m edges such that cr(G′) ≥ (1/28 - o(1))cr(G). © Springer-Verlag Berlin Heidelberg 2007.