On the connectedness of some geometric graphs

Joseph Zaks

doi:10.1016/0095-8956(90)90023-S

On the connectedness of some geometric graphs

Joseph Zaks

Department of Mathematics

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

Abstract

Fos S ⊆ R, let G(S^d) denote the graph, obtained from all the points of the real d-space R^d having their coordinates in S, by connecting every two points which are at distance one. If S = Q (the rationals), or Q[√k], or Q[a] where a is a real algebraic number, then G(S^d) is shown to be connected for all large values of d. Similar results are obtained in the case of S = Q[ⁿ k], where the distance is computed in the l_p-norm, p an integer, p ≥ 3 and n∥p (or n = 2).

Original language	English
Pages (from-to)	143-150
Number of pages	8
Journal	Journal of Combinatorial Theory. Series B
Volume	49
Issue number	2
DOIs	https://doi.org/10.1016/0095-8956(90)90023-S
State	Published - Aug 1990

ASJC Scopus subject areas

Theoretical Computer Science
Discrete Mathematics and Combinatorics
Computational Theory and Mathematics

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10.1016/0095-8956(90)90023-S

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title = "On the connectedness of some geometric graphs",

abstract = "Fos S ⊆ R, let G(Sd) denote the graph, obtained from all the points of the real d-space Rd having their coordinates in S, by connecting every two points which are at distance one. If S = Q (the rationals), or Q[√k], or Q[a] where a is a real algebraic number, then G(Sd) is shown to be connected for all large values of d. Similar results are obtained in the case of S = Q[n k], where the distance is computed in the lp-norm, p an integer, p ≥ 3 and n∥p (or n = 2).",

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AB - Fos S ⊆ R, let G(Sd) denote the graph, obtained from all the points of the real d-space Rd having their coordinates in S, by connecting every two points which are at distance one. If S = Q (the rationals), or Q[√k], or Q[a] where a is a real algebraic number, then G(Sd) is shown to be connected for all large values of d. Similar results are obtained in the case of S = Q[n k], where the distance is computed in the lp-norm, p an integer, p ≥ 3 and n∥p (or n = 2).

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ER -

On the connectedness of some geometric graphs

Abstract

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