Graphs which are locally a cube
type: Item
instance of: scholarly article
title: Graphs which are locally a cube
zbMATH Open document ID: 0532.05050
DOI: 10.1016/0012-365X(83)90116-4
published in: Discrete Mathematics
publication date: 1983-01-01T00:00:00Z
review text: A graph G is said to be locally a cube iff each vertex of G has a neighborhood isomorphic to the 1-skeleton of the 3-dimensional cube. It is shown that the only graphs that are locally cubes are the 24-cell and the complement of the (3\(\times 5)\)-grid. The vertices of the 24-cell are the vectors \(+e_ i\pm e_ j(i\neq j) of{\mathbb{R}}^ 4\) where \(\{e_ i,e_ 2,e_ 3,e_ 4\}\) is the standard basis. Two vertices are adjacent if the angle between corresponding vectors is 60\(\circ\). Th\(e(p\times q)\)-grid is the graph whose vertices are the pq ordered pairs (i,j) with \(i=1,...,p\) and \(j=1,...,q\), two vertices being adjacent iff they have one coordinate in common. The author reports that A. Brouwer also had obtained this result (unpublished) and has characterized those graphs which are locally the complement of a (\(p\times q)\)-grid.
Mathematics Subject Classification ID: 05C75
Mathematics Subject Classification ID: 05C99
zbMATH DE Number: 3843799
zbMATH Keywords: neighborhood
zbMATH Keywords: 1-skeleton of the 3-dimensional cube
zbMATH Keywords: (\(p\times q)\)-grid
author: Dominique Buset
reviewed by: R. C. Entringer
MaRDI profile type: Publication
cites work: Q5672665
cites work: Locally petersen graphs
description: scientific article; zbMATH DE number 3843799