### Abstract

Pinchasi and Radoičić [On the number of edges in geometric graphs with no self-intersecting cycle of length 4, in: J. Pach (Ed.), Towards a Theory of Geometric Graphs, Contemporary Mathematics, vol. 342, American Mathematical Society, Providence, RI, 2004] used the following observation to bound the number of edges of a topological graph without a self-crossing cycle of length 4: if we make a list of the neighbors for every vertex in such a graph and order these lists cyclically according to the order of the emanating edges, then the common elements in any two lists have reversed cyclic order. Building on their work we give an improved estimate on the size of the lists having this property. As a consequence we get that a topological graph on n vertices not containing a self-crossing C_{4} has O(n^{3/2} log n) edges. Our result also implies that n pseudo-circles in the plane can be cut into O(n^{3/2} log n) pseudo-segments, which in turn implies bounds on point-curve incidences and on the complexity of a level of an arrangement of curves.

Original language | English |
---|---|

Pages (from-to) | 675-691 |

Number of pages | 17 |

Journal | Journal of Combinatorial Theory, Series A |

Volume | 113 |

Issue number | 4 |

DOIs | |

Publication status | Published - May 2006 |

### Fingerprint

### Keywords

- Cyclic order
- Extremal combinatorics
- Pseudocircles
- Topological graphs

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Journal of Combinatorial Theory, Series A*,

*113*(4), 675-691. https://doi.org/10.1016/j.jcta.2005.07.002

**Intersection reverse sequences and geometric applications.** / Marcus, Adam; Tardos, G.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory, Series A*, vol. 113, no. 4, pp. 675-691. https://doi.org/10.1016/j.jcta.2005.07.002

}

TY - JOUR

T1 - Intersection reverse sequences and geometric applications

AU - Marcus, Adam

AU - Tardos, G.

PY - 2006/5

Y1 - 2006/5

N2 - Pinchasi and Radoičić [On the number of edges in geometric graphs with no self-intersecting cycle of length 4, in: J. Pach (Ed.), Towards a Theory of Geometric Graphs, Contemporary Mathematics, vol. 342, American Mathematical Society, Providence, RI, 2004] used the following observation to bound the number of edges of a topological graph without a self-crossing cycle of length 4: if we make a list of the neighbors for every vertex in such a graph and order these lists cyclically according to the order of the emanating edges, then the common elements in any two lists have reversed cyclic order. Building on their work we give an improved estimate on the size of the lists having this property. As a consequence we get that a topological graph on n vertices not containing a self-crossing C4 has O(n3/2 log n) edges. Our result also implies that n pseudo-circles in the plane can be cut into O(n3/2 log n) pseudo-segments, which in turn implies bounds on point-curve incidences and on the complexity of a level of an arrangement of curves.

AB - Pinchasi and Radoičić [On the number of edges in geometric graphs with no self-intersecting cycle of length 4, in: J. Pach (Ed.), Towards a Theory of Geometric Graphs, Contemporary Mathematics, vol. 342, American Mathematical Society, Providence, RI, 2004] used the following observation to bound the number of edges of a topological graph without a self-crossing cycle of length 4: if we make a list of the neighbors for every vertex in such a graph and order these lists cyclically according to the order of the emanating edges, then the common elements in any two lists have reversed cyclic order. Building on their work we give an improved estimate on the size of the lists having this property. As a consequence we get that a topological graph on n vertices not containing a self-crossing C4 has O(n3/2 log n) edges. Our result also implies that n pseudo-circles in the plane can be cut into O(n3/2 log n) pseudo-segments, which in turn implies bounds on point-curve incidences and on the complexity of a level of an arrangement of curves.

KW - Cyclic order

KW - Extremal combinatorics

KW - Pseudocircles

KW - Topological graphs

UR - http://www.scopus.com/inward/record.url?scp=33645550027&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33645550027&partnerID=8YFLogxK

U2 - 10.1016/j.jcta.2005.07.002

DO - 10.1016/j.jcta.2005.07.002

M3 - Article

AN - SCOPUS:33645550027

VL - 113

SP - 675

EP - 691

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

SN - 0097-3165

IS - 4

ER -