Let G be the graph obtained from all the rational points in the d-space Ed by connecting every pair at Euclidean distance one. It is known that G is a connected graph, provided d ≥ 5. We establish an inequality of the form distG(x, y) ≤ ⌈|x - y|⌉ + 1, for all d ≥ 8, between the Euclidean distance |x - y| of any two rational points x and y and their corresponding distance distG(x, y) in the graph G. A slightly weaker relation is shown to hold in dimensions 5, 6 and 7.
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics