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Graphs which are locally a cube. (English) Zbl 0532.05050
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 The author reports that A. Brouwer also had obtained this result (unpublished) and has characterized those graphs which are locally the complement of a (\(p\times q)\)-grid.
Reviewer: R.C.Entringer

MSC:
05C75 Structural characterization of families of graphs
05C99 Graph theory
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References:
[1] A. Brouwer, Personal communication.
[2] Coxeter, H.S.M., Regular polytopes, (1973), Dover Publications New York · Zbl 0258.05119
[3] Hall, J.I., Locally Petersen graphs, J. graph theory, 4, 173-187, (1980) · Zbl 0407.05041
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