Abstract
For a setS of points in the plane, letd 1>d 2>... 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|≥constk 2. IfS consists of the vertices of a convex polygon and |S|≥constk 2, 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.
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Erdős, P., Lovász, L. & Vesztergombi, K. On the graph of large distances. Discrete Comput Geom 4, 541–549 (1989). https://doi.org/10.1007/BF02187746
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DOI: https://doi.org/10.1007/BF02187746