On the graph of large distances

P. Erdős, L. Lovász, K. Vesztergombi

Research output: Contribution to journalArticle

11 Citations (Scopus)

Abstract

For a set S of points in the plane, let d1>d2>... denote the different distances determined by S. Consider the graph G(S, k) whose vertices are the elements of S, and two are joined by an edge iff their distance is at least dk. It is proved that the chromatic number of G(S, k) is at most 7 if |S|≥const k2. If S consists of the vertices of a convex polygon and |S|≥const k2, then the chromatic number of G(S, k) is at most 3. Both bounds are best possible. If S consists of the vertices of a convex polygon then G(S, k) has a vertex of degree at most 3 k - 1. This implies that in this case the chromatic number of G(S, k) is at most 3 k. The best bound here is probably 2 k+1, which is tight for the regular (2 k+1)-gon.

Original languageEnglish
Pages (from-to)541-549
Number of pages9
JournalDiscrete & Computational Geometry
Volume4
Issue number1
DOIs
Publication statusPublished - Dec 1989

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ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

Cite this

On the graph of large distances. / Erdős, P.; Lovász, L.; Vesztergombi, K.

In: Discrete & Computational Geometry, Vol. 4, No. 1, 12.1989, p. 541-549.

Research output: Contribution to journalArticle

Erdős, P. ; Lovász, L. ; Vesztergombi, K. / On the graph of large distances. In: Discrete & Computational Geometry. 1989 ; Vol. 4, No. 1. pp. 541-549.
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