Geometric Spanners With Small Chromatic Number

TL;DR

This paper investigates the minimum stretch factor for geometric spanners with bounded chromatic number, providing exact values for small k, bounds for larger k, and an online variant analysis.

Contribution

It determines exact stretch factors for small chromatic numbers and bounds for larger ones, including an online setting, advancing understanding of chromatic geometric spanners.

Findings

t(2)=3, t(3)=2, t(4)=√2 for the offline case

Existence of near-optimal spanners with O(|P|) edges and bounded chromatic number

Exact values for t(k) in the online variant for small k

Abstract

Given an integer $k \geq 2$, 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 $\epsilon >0$, there exists a $(1+\epsilon)t(k)$-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 $k>4$.