Lecture Notes in Computer Science Volume 4372, 2007, pp 174-183

On the Decay of Crossing Numbers

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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 e such that \({\rm cr}(G-e) \geq k-c\sqrt{k}\) . They showed only that G always has an edge e with \({\rm cr}(G-e) \geq \frac{2}{5}{\rm cr}(G)-O(1)\) . 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 \((1-\frac{1}{24\epsilon})m\) edges such that \({\rm cr}(G') \geq (\frac{1}{28}-o(1)){\rm cr}(G)\) .