Embedding products of graphs into Euclidean spaces

Details Embedding products of graphs into Euclidean spaces Embedding products of graphs into Euclidean spaces

2008-08-08

arxiv.org/abs/0808.1199v1

Fundamenta Mathematicae 179 (2003), p. 191--197

Mathematics

Geometric Topology

in English and in Russian, 5 pages, 2 figures

Scientific paper

For any collection of graphs we find the minimal dimension d such that the product of these graphs is embeddable into the d-dimensional Euclidean space. In particular, we prove that the n-th powers of the Kuratowsky graphs are not embeddable into the 2n-dimensional Euclidean space. This is a solution of a problem of Menger from 1929. The idea of the proof is the reduction to a problem from so-called Ramsey link theory: we show that any embedding LkO into the (2n-1)-sphere, where O is a vertex of a Kuratowsky graph, has a pair of linked (n-1)-spheres.

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