Abstract
Twenty years ago, Ajtai et al. and, independently, Leighton discovered that the crossing number of any graph with v vertices and e > 4v edges is at least ce3/v2, where c > 0 is an absolute constant. This result, known as the "Crossing Lemma," has found many important applications in discrete and computational geometry. It is tight up to a multiplicative constant. Here we improve the best known value of the constant by showing that the result holds with c > 1024/31827 > 0.032. The proof has two new ingredients, interesting in their own right. We show that (1) if a graph can be drawn in the plane so that every edge crosses at most three others, then its number of edges cannot exceed 5.5(v-2); and (2) the crossing number of any graph is at least \(\frac73e-\frac{25}3(v-2).\) Both bounds are tight up to an additive constant (the latter one in the range \(4v\le e\le 5v\)).
Article PDF
Similar content being viewed by others
Author information
Authors and Affiliations
Corresponding authors
Rights and permissions
About this article
Cite this article
Pach, J., Radoicic, R., Tardos, G. et al. Improving the Crossing Lemma by Finding More Crossings in Sparse Graphs. Discrete Comput Geom 36, 527–552 (2006). https://doi.org/10.1007/s00454-006-1264-9
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00454-006-1264-9