### 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 |
---|---|

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 2007 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Journal of Combinatorial Theory, Series A*,

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

**On the maximum number of edges in quasi-planar graphs.** / Ackerman, Eyal; Tardos, G.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory, Series A*, vol. 114, no. 3, pp. 563-571. https://doi.org/10.1016/j.jcta.2006.08.002

}

TY - JOUR

T1 - On the maximum number of edges in quasi-planar graphs

AU - Ackerman, Eyal

AU - Tardos, G.

PY - 2007/4

Y1 - 2007/4

N2 - 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.

AB - 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.

KW - Discharging method

KW - Geometric graphs

KW - Quasi-planar graphs

KW - Topological graphs

KW - Turán-type problems

UR - http://www.scopus.com/inward/record.url?scp=33846817022&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33846817022&partnerID=8YFLogxK

U2 - 10.1016/j.jcta.2006.08.002

DO - 10.1016/j.jcta.2006.08.002

M3 - Article

AN - SCOPUS:33846817022

VL - 114

SP - 563

EP - 571

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

SN - 0097-3165

IS - 3

ER -