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Discrete & Computational Geometry

, Volume 4, Issue 4, pp 349–364 | Cite as

Embeddings of graphs in euclidean spaces

  • J. Reiterman
  • V. Rödl
  • E. Šiňajová
Article

Abstract

The dimension of a graphG=(V, E) is the minimum numberd such that there exists a representation\(x \to \bar x \in R^d (x \in V)\) and a thresholdt such thatxy εE iff\(\mathop x\limits^ - \mathop y\limits^ - \geqslant t\). We prove that d(G)≤n−x(G) and\(d(G) \leqslant n - \sqrt n \) wheren=|V| andx(G) is the chromatic number ofG; we present upper bounds for the dimension of graphs with a large girth and we show that the complement of a forest can be represented by unit vectors inR6. We prove that d(G)≥1/15n for most graphs and that there are 3-regular graphs with d(G)≥c logn/log logn.

Keywords

Euclidean Space Bipartite Graph Chromatic Number Discrete Comput Geom Stochastic Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York Inc 1989

Authors and Affiliations

  • J. Reiterman
    • 1
  • V. Rödl
    • 1
  • E. Šiňajová
    • 1
  1. 1.Department of Mathematics FJFITechnical University of PraguePrague 1Czechoslovakia

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