On the graph of large distances

Abstract

For a set S of points in the plane, let d 1 >d 2 >... 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 d k . It is proved that the chromatic number of G(S, k) is at most 7 if |S|≥const k 2 . If S consists of the vertices of a convex polygon and |S|≥const k 2 , 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. © 1989 Springer-Verlag New York Inc.