We prove that the crossing number of a graph decays in a "continuous fashion" in the following sense. For any ? > 0 there is a ? > 0 such that for a sufficiently large n, every graph G with n vertices and m ? n 1+? edges, has a subgraph G? of at most (1 ? ?)m edges and crossing number at least (1 ? ?)CR(G). This generalizes the result of J. Fox and Cs. Toth.
['Jakub Černý', 'Jan Kynčl', 'Géza Tóth']
Improvement on the decay of crossing numbers