Discrete & Computational Geometry

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

On the graph of large distances

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


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.


Chromatic Number Discrete Comput Geom Convex Polygon Convex Case Counterclockwise Order 
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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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