### Abstract

A topological graph is quasi-planar, if it does not contain three pairwise crossing edges. Agarwal et al. [P.K. Agarwal, B. Aronov, J. Pach, R. Pollack, M. Sharir, Quasi-planar graphs have a linear number of edges, Combinatorica 17 (1) (1997) 1-9] proved that these graphs have a linear number of edges. We give a simple proof for this fact that yields the better upper bound of 8n edges for n vertices. Our best construction with 7 n - O (1) edges comes very close to this bound. Moreover, we show matching upper and lower bounds for several relaxations and restrictions of this problem. In particular, we show that the maximum number of edges of a simple quasi-planar topological graph (i.e., every pair of edges have at most one point in common) is 6.5 n - O (1), thereby exhibiting that non-simple quasi-planar graphs may have many more edges than simple ones.

Original language | English |
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Pages (from-to) | 563-571 |

Number of pages | 9 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 114 |

Issue number | 3 |

DOIs | |

Publication status | Published - Apr 1 2007 |

### Keywords

- Discharging method
- Geometric graphs
- Quasi-planar graphs
- Topological graphs
- Turán-type problems

### ASJC Scopus subject areas

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

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## Cite this

*Journal of Combinatorial Theory. Series A*,

*114*(3), 563-571. https://doi.org/10.1016/j.jcta.2006.08.002