Improvement on the Decay of Crossing Numbers.

We prove that the crossing number of a graph decays in a 'continuous fashion' in the following sense. For any ε > 0 there is a δ > 0 such that for a sufficiently large n, every graph G with n vertices and m ≥ n edges, has a subgraph G′ of at most (1 − δ) m edges and crossing number at least (1 − ε)CR( G). This generalizes the result of J. Fox and Cs. Tóth. [ABSTRACT FROM AUTHOR]