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 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 \({\rm cr}(G') \geq (\frac{1}{28}-o(1)){\rm cr}(G)\).
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Fox, J., Tóth, C.D. (2007). On the Decay of Crossing Numbers. In: Kaufmann, M., Wagner, D. (eds) Graph Drawing. GD 2006. Lecture Notes in Computer Science, vol 4372. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70904-6_18
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DOI: https://doi.org/10.1007/978-3-540-70904-6_18
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