Embedding graphs in Euclidean space

Frankl, Nóra, Kupavskii, Andrey and Swanepoel, Konrad ORCID: 0000-0002-1668-887X (2020) Embedding graphs in Euclidean space. Journal of Combinatorial Theory, Series A, 171. ISSN 0097-3165

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Abstract

The dimension of a graph G is the smallest d for which its vertices can be embedded in d-dimensional Euclidean space in the sense that the distances between endpoints of edges equal 1 (but there may be other unit distances). Answering a question of Erdős and Simonovits (1980) [5], we show that any graph with less than (d+22) edges has dimension at most d. Improving their result, we prove that the dimension of a graph with maximum degree d is at most d. We show the following Ramsey result: if each edge of the complete graph on 2d vertices is coloured red or blue, then either the red graph or the blue graph can be embedded in Euclidean d-space. We also derive analogous results for embeddings of graphs into the (d−1)-dimensional sphere of radius 1/2.

Item Type:	Article
Official URL:	https://www.journals.elsevier.com/journal-of-combi...
Additional Information:	© 2019 Elsevier B.V.
Divisions:	Mathematics
Subjects:	Q Science > QA Mathematics
Date Deposited:	26 Sep 2019 10:06
Last Modified:	11 Sep 2025 10:02
URI:	http://eprints.lse.ac.uk/id/eprint/101727

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