Advertisement

Discrete & Computational Geometry

, Volume 36, Issue 4, pp 527–552 | Cite as

Improving the Crossing Lemma by Finding More Crossings in Sparse Graphs

  • Janos  Pach
  • Rados Radoicic
  • Gabor Tardos
  • Geza Toth
Article

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 \(4v\le e\le 5v\)).

Keywords

Additive Constant Parallel Edge Triangular Face Simple Face Strong Neighbor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Copyright information

© Springer 2006

Authors and Affiliations

  1. 1.Courant Institute, N.Y.U., 251 Mercer StreetNew York, NY 10012USA
  2. 2.Renyi Institute of Mathematics, Hungarian Academy of Sciences, Pf. 127, H-1364BudapestHungary
  3. 3.Department of Mathematics, Baruch College, CUNY, One Bernard Baruch WayNew York, NY 10010USA

Personalised recommendations