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ON THE UPPER CHROMATIC NUMBERS OF THE AARON F. ARCHER
 

Summary: ON THE UPPER CHROMATIC NUMBERS OF THE
REALS
AARON F. ARCHER
Abstract. Let S be a metric space and let k be a positive in
teger. Define
(k) (S) to be the smallest positive integer m such
that for every k \Theta m array D = (D ij ) of positive real numbers, S
can be colored with the colors 1; 2; : : : ; m such that no two points
of distance D ij are both colored j. We improve the best upper
bound known on
(k) (R) from 32 k k! to d4eke, where e is the base
of the natural logarithm. We prove a conjecture of A. Abrams
that
(k) (Z) =
(k) (R) for all k 2 N, extend this result to higher
dimensions under the l 1 and l 1 norms, and prove that the upper
chromatic numbers are finite for these spaces. We also introduce
a new related chromatic quantity of a graph G, the chromatic ca
pacity, cap (G).
1. Introduction

  

Source: Archer, Aaron - Algorithms and Optimization Group, AT&T Labs-Research

 

Collections: Computer Technologies and Information Sciences