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On diversity and stability of unit bases for the Euclidean metric

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Abstract

We prove the existence of a family Ω(n) of 2c (where c is the cardinality of the continuum) subgraphs of the unit distance graph (E n, 1) of the Euclidean space E n, n ≥ 2, such that (a) for each graph G ϵ Ω(n), any homomorphism of G to (E n, 1) is an isometry of E n; moreover, for each subgraph G 0 of the graph G obtained from G by deleting less than c vertices, less than c stars, and less than c edges (we call such a subgraph reduced), any homomorphism of G 0 to (E n, 1) is an isometry (of the set of the vertices of G 0); (b) each graph G ϵ Ω(n) cannot be homomorphically mapped to any other graph of the family Ω(n), and the same is true for each reduced subgraph of G.

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  1. Department of Mathematics, Purdue University, West Lafayette, 2501 Soldiers Home Rd. Apt. 6F, West Lafayette, IN, 47906, USA

    Alexandr Kuzminykh

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  1. Alexandr Kuzminykh
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Correspondence to Alexandr Kuzminykh.

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Kuzminykh, A. On diversity and stability of unit bases for the Euclidean metric. J. Geom. 94, 143–150 (2009). https://doi.org/10.1007/s00022-009-0010-x

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  • DOI: https://doi.org/10.1007/s00022-009-0010-x

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