Improving the Crossing Lemma by finding more crossings in sparse graphs

Janos Pach, Rados Radoicic, Gabor Tardos, Geza Toth

Research output: Contribution to journalArticle

57 Citations (Scopus)


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 7/3e-25/3(v-2). Both bounds are tight up to an additive constant (the latter one in the range 4v ≤ e ≤ 5v).

Original languageEnglish
Pages (from-to)527-552
Number of pages26
JournalDiscrete and Computational Geometry
Issue number4
Publication statusPublished - Dec 2006

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

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