### Abstract

Given a point set P in the plane, the Delaunay graph with respect to axis-parallel rectangles is a graph defined on the vertex set P, whose two points p, q εP are connected by an edge if and only if there is a rectangle parallel to the coordinate axes that contains p and q, but no other elements of P. The following question of Even et al. (SIAM J Comput 33 (2003) 94-136) was motivated by a frequency assignment problem in cellular telephone networks: Does there exist a constant c > 0 such that the Delaunay graph of any set of n points in general position in the plane contains an independent set of size at least cn? We answer this question in the negative, by proving that the largest independent set in a randomly and uniformly selected point set in the unit square is O(n log ^{2} log n/log n), with probability tending to 1. We also show that our bound is not far from optimal, as the Delaunay graph of a uniform random set of n points almost surely has an independent set of size at least cn log log n/(log n log log log n). We give two further applications of our methods: (1) We construct two-dimensional n-element partially ordered sets such that the size of the largest independent sets of vertices in their Hasse diagrams is o(n). This answers a question of Matoušek and Přívětivý (Combinat Probab Comput 15 (2006) 473-475) and improves a result of Kříž and Nešetřil (Order 8 (1991) 41-48). (2) For any positive integers c and d, we prove the existence of a planar point set with the property that no matter how we color its elements by c colors, we find an axis-parallel rectangle containing at least d points, all of which have the same color. This solves an old problem from the work of Brass et al. (Research Problem in Discrete Geometry Springer-Verlag, New York, 2005).

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

Pages (from-to) | 11-23 |

Number of pages | 13 |

Journal | Random Structures and Algorithms |

Volume | 34 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 2009 |

### Fingerprint

### Keywords

- Delauney graphs
- Frequency assignment
- Graph coloring
- Hasse diagram
- Voronoi diagrams

### ASJC Scopus subject areas

- Computer Graphics and Computer-Aided Design
- Software
- Mathematics(all)
- Applied Mathematics

### Cite this

*Random Structures and Algorithms*,

*34*(1), 11-23. https://doi.org/10.1002/rsa.20246

**Delaunay graphs of point sets in the plane with respect to axis-parallel rectangles.** / Chen, Xiaomin; Pach, János; Szegedy, Mario; Tardos, G.

Research output: Contribution to journal › Article

*Random Structures and Algorithms*, vol. 34, no. 1, pp. 11-23. https://doi.org/10.1002/rsa.20246

}

TY - JOUR

T1 - Delaunay graphs of point sets in the plane with respect to axis-parallel rectangles

AU - Chen, Xiaomin

AU - Pach, János

AU - Szegedy, Mario

AU - Tardos, G.

PY - 2009/1

Y1 - 2009/1

N2 - Given a point set P in the plane, the Delaunay graph with respect to axis-parallel rectangles is a graph defined on the vertex set P, whose two points p, q εP are connected by an edge if and only if there is a rectangle parallel to the coordinate axes that contains p and q, but no other elements of P. The following question of Even et al. (SIAM J Comput 33 (2003) 94-136) was motivated by a frequency assignment problem in cellular telephone networks: Does there exist a constant c > 0 such that the Delaunay graph of any set of n points in general position in the plane contains an independent set of size at least cn? We answer this question in the negative, by proving that the largest independent set in a randomly and uniformly selected point set in the unit square is O(n log 2 log n/log n), with probability tending to 1. We also show that our bound is not far from optimal, as the Delaunay graph of a uniform random set of n points almost surely has an independent set of size at least cn log log n/(log n log log log n). We give two further applications of our methods: (1) We construct two-dimensional n-element partially ordered sets such that the size of the largest independent sets of vertices in their Hasse diagrams is o(n). This answers a question of Matoušek and Přívětivý (Combinat Probab Comput 15 (2006) 473-475) and improves a result of Kříž and Nešetřil (Order 8 (1991) 41-48). (2) For any positive integers c and d, we prove the existence of a planar point set with the property that no matter how we color its elements by c colors, we find an axis-parallel rectangle containing at least d points, all of which have the same color. This solves an old problem from the work of Brass et al. (Research Problem in Discrete Geometry Springer-Verlag, New York, 2005).

AB - Given a point set P in the plane, the Delaunay graph with respect to axis-parallel rectangles is a graph defined on the vertex set P, whose two points p, q εP are connected by an edge if and only if there is a rectangle parallel to the coordinate axes that contains p and q, but no other elements of P. The following question of Even et al. (SIAM J Comput 33 (2003) 94-136) was motivated by a frequency assignment problem in cellular telephone networks: Does there exist a constant c > 0 such that the Delaunay graph of any set of n points in general position in the plane contains an independent set of size at least cn? We answer this question in the negative, by proving that the largest independent set in a randomly and uniformly selected point set in the unit square is O(n log 2 log n/log n), with probability tending to 1. We also show that our bound is not far from optimal, as the Delaunay graph of a uniform random set of n points almost surely has an independent set of size at least cn log log n/(log n log log log n). We give two further applications of our methods: (1) We construct two-dimensional n-element partially ordered sets such that the size of the largest independent sets of vertices in their Hasse diagrams is o(n). This answers a question of Matoušek and Přívětivý (Combinat Probab Comput 15 (2006) 473-475) and improves a result of Kříž and Nešetřil (Order 8 (1991) 41-48). (2) For any positive integers c and d, we prove the existence of a planar point set with the property that no matter how we color its elements by c colors, we find an axis-parallel rectangle containing at least d points, all of which have the same color. This solves an old problem from the work of Brass et al. (Research Problem in Discrete Geometry Springer-Verlag, New York, 2005).

KW - Delauney graphs

KW - Frequency assignment

KW - Graph coloring

KW - Hasse diagram

KW - Voronoi diagrams

UR - http://www.scopus.com/inward/record.url?scp=60349088927&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=60349088927&partnerID=8YFLogxK

U2 - 10.1002/rsa.20246

DO - 10.1002/rsa.20246

M3 - Article

AN - SCOPUS:60349088927

VL - 34

SP - 11

EP - 23

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

SN - 1042-9832

IS - 1

ER -