## Abstract

A topological graph G is a graph drawn in the plane with vertices represented by points and edges represented by continuous arcs connecting the vertices. If every edge is drawn as a straight-line segment, then G is called a geometric graph. A k-grid in a topological graph is a pair of subsets of the edge set, each of size k, such that every edge in one subset crosses every edge in the other subset. It is known that every n-vertex topological graph with no k-grid has ^{Ok}(n) edges. We conjecture that the number of edges of every n-vertex topological graph with no k-grid such that all of its 2k edges have distinct endpoints is ^{Ok}(n). This conjecture is shown to be true apart from an iterated logarithmic factor log.

Original language | English |
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Pages (from-to) | 710-723 |

Number of pages | 14 |

Journal | Computational Geometry: Theory and Applications |

Volume | 47 |

Issue number | 7 |

DOIs | |

State | Published - Aug 2014 |

## Keywords

- Geometric graphs
- Topological graphs
- Turán-type problems

## ASJC Scopus subject areas

- Computer Science Applications
- Geometry and Topology
- Control and Optimization
- Computational Theory and Mathematics
- Computational Mathematics