Geometric spanners with small chromatic number

摘要

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) = root 2, and give upper and lower bounds on t(k) for k > 4. We also show that for any is an element of > 0, there exists a (1 + is an element of)t(k)-spanner for P that has 0 (|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 + root 3, t(4) = 1 + root 2, and give upper and lower bounds on t(k) for k > 4. (C) 2008 Elsevier B.V. All rights reserved.