Discrete & Computational Geometry

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

Improving the Crossing Lemma by Finding More Crossings in Sparse Graphs

  • Janos  PachEmail author
  • Rados RadoicicEmail author
  • Gabor TardosEmail author
  • Geza TothEmail author


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


Additive Constant Parallel Edge Triangular Face Simple Face Strong Neighbor 
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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

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