On the connectedness of some geometric graphs
type: Item
instance of: scholarly article
title: On the connectedness of some geometric graphs
zbMATH Open document ID: 0728.05039
DOI: 10.1016/0095-8956(90)90023-S
author: Joseph Zaks
published in: Journal of Combinatorial Theory. Series B
publication date: 1990-01-01T00:00:00Z
review text: Let \(S\subseteq R\). Then by \(G(S^ d)\) we denote a graph whose vertices are all the points of the real space \(R^ d\) with their coordinates in S and two vertices are connected by an edge if their distance is 1. In the paper it is proved that G(Q(\(\sqrt{k})^ d)\) is connected for all \(k\in N\), \(d\geq 5\); G(Q(\(\sqrt{k})^ 2)\) has countably infinitely many connected components for \(k=1,2,3,5,6(mod 8)\), and that for every algebraic number a, \(G(Q(a)^ d)\) is a connected graph for every large d.
Mathematics Subject Classification ID: 05C40
zbMATH DE Number: 4202296
zbMATH Keywords: connectivity
zbMATH Keywords: geometric graph
reviewed by: Peter Horák
MaRDI profile type: Publication
cites work: Unit-distance graphs in rational n-spaces
cites work: Q3259107
cites work: Coloring abelian groups
cites work: Distances realized by sets covering the plane
description: scientific article; zbMATH DE number 4202296