Geometric Spanners with Small Chromatic Number

Abstract

Given an integer k ≥ 2, we consider the problem of computing the smallest real number t(k) such that for each set P of points in the plane, there exists a t(k)-spanner for P that has chromatic number at most k. We prove that t(2) = 3, t(3) = 2, $t(4) = \sqrt{2}$, and give upper and lower bounds on t(k) for k > 4. We also show that for any ε> 0, there exists a (1 + ε)t(k)-spanner for P that has O(|P|) edges and chromatic number at most k. Finally, we consider an on-line variant of the problem where the points of P are given one after another, and the color of a point must be assigned at the moment the point is given. In this setting, we prove that t(2) = 3, $t(3) = 1+ \sqrt{3}$, $t(4) = 1+ \sqrt{2}$, and give upper and lower bounds on t(k) for k > 4.

Research partially supported by HPCVL, NSERC, MRI, CFI, and MITACS.