Discrete & Computational Geometry

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

Embeddings of graphs in euclidean spaces

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


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.


Euclidean Space Bipartite Graph Chromatic Number Discrete Comput Geom Stochastic Matrix 
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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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