On the Degenerate Crossing Number

Eyal Ackerman, Rom Pinchasi

Research output: Contribution to journalArticlepeer-review


The degenerate crossing number cr*(G) of a graph G is the minimum number of crossing points of edges in any drawing of G as a simple topological graph in the plane. This notion was introduced by Pach and Tóth who showed that for a graph G with n vertices and e ≥ 4n edges cr*(G)=Ω(e4 / n4). In this paper we completely resolve the main open question about degenerate crossing numbers and show that cr*(G)=Ω (e3 / n2), provided that e≥ 4n. This bound is best possible (apart for the multiplicative constant) as it matches the tight lower bound for the standard crossing number of a graph.

Original languageEnglish
Pages (from-to)695-702
Number of pages8
JournalDiscrete and Computational Geometry
Issue number3
StatePublished - Apr 2013

Bibliographical note

Funding Information:
Rom Pinchasi was supported by ISF Grant (Grant No. 1357/12) and by BSF Grant (Grant No. 2008290).


  • Crossing Lemma
  • Crossing number
  • Simple topological graphs

ASJC Scopus subject areas

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


Dive into the research topics of 'On the Degenerate Crossing Number'. Together they form a unique fingerprint.

Cite this