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Discrete & Computational Geometry

, Volume 4, Issue 6, pp 541–549 | Cite as

On the graph of large distances

  • P. Erdős
  • L. Lovász
  • K. Vesztergombi
Article

Abstract

For a setS of points in the plane, letd1>d2>... denote the different distances determined byS. Consider the graphG(S, k) whose vertices are the elements ofS, and two are joined by an edge iff their distance is at leastd k . It is proved that the chromatic number ofG(S, k) is at most 7 if |S|≥constk2. IfS consists of the vertices of a convex polygon and |S|≥constk2, then the chromatic number ofG(S, k) is at most 3. Both bounds are best possible. IfS consists of the vertices of a convex polygon thenG(S, k) has a vertex of degree at most 3k − 1. This implies that in this case the chromatic number ofG(S, k) is at most 3k. The best bound here is probably 2k+1, which is tight for the regular (2k+1)-gon.

Keywords

Chromatic Number Discrete Comput Geom Convex Polygon Convex Case Counterclockwise Order 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York Inc. 1989

Authors and Affiliations

  • P. Erdős
    • 1
  • L. Lovász
    • 2
  • K. Vesztergombi
    • 3
  1. 1.Mathematical Research InstituteHungarian Academy of SciencesBudapestHungary
  2. 2.Department of Computer ScienceEötvös Loránd UniversityBudapestHungary
  3. 3.Department of Mathematics, Faculty of Electrical EngineeringBudapest University of TechnologyBudapestHungary

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