Geometric spanners with small chromatic number

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, , 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, , , and give upper and lower bounds on t(k) for k∈>∈4.