### 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 ce^{3}/v^{2}, 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 language | English |
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Pages (from-to) | 527-552 |

Number of pages | 26 |

Journal | Discrete and Computational Geometry |

Volume | 36 |

Issue number | 4 |

DOIs | |

Publication status | Published - Dec 2006 |

### ASJC Scopus subject areas

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

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## Cite this

*Discrete and Computational Geometry*,

*36*(4), 527-552. https://doi.org/10.1007/s00454-006-1264-9