Distance graph on [formula omitted] with ℓ 1 norm

Abstract

A long-standing open problem in combinatorial geometry is the chromatic number of the unit-distance graph in R n ; here points are adjacent if their distance in the ℓ 2 norm is 1. For n=2, we know the answer is between 4 and 7. Little is known about other dimensions. The subgraphs induced by the rational points have been studied with limited success in small dimensions. We consider the analogous problem on the n-dimensional integer grid with fixed distance in the ℓ 1 norm. That is, we make two integer grid points adjacent if the sum of the absolute differences in their coordinate values is r. Let the chromatic number of this graph be χ( Z,r) . The main results of this paper are (i) χ( Z n,2)=2n for all n, and (ii) (1.139) n⩽χ( Z n,r)⩽(1/ 2πn )(5 e) n for all n and even r. We also give bounds useful for small values of n and r. We also consider the lower and upper bounds on the n-dimensional real space with unit distance under ℓ p norm for 1⩽ p⩽∞.