Uniform distances in rational unit-distance graphs

Joseph Zaks

doi:10.1016/0012-365X(92)90301-U

Uniform distances in rational unit-distance graphs

Joseph Zaks

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Let G be the graph obtained from all the rational points in the d-space E^d 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 dist_G(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 dist_G(x, y) in the graph G. A slightly weaker relation is shown to hold in dimensions 5, 6 and 7.

Original language	English
Pages (from-to)	307-311
Number of pages	5
Journal	Discrete Mathematics
Volume	109
Issue number	1-3
DOIs	https://doi.org/10.1016/0012-365X(92)90301-U
State	Published - 12 Nov 1992

ASJC Scopus subject areas

Theoretical Computer Science
Discrete Mathematics and Combinatorics

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10.1016/0012-365X(92)90301-U

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title = "Uniform distances in rational unit-distance graphs",

abstract = "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.",

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AB - 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.

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Uniform distances in rational unit-distance graphs

Abstract

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