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Combinatorica

, Volume 16, Issue 1, pp 59–85 | Cite as

Graphs of small dimensions

  • Nancy Eaton
  • Vojtěch Rödl
Article

Abstract

LetG=(V, E) be a graph withn vertices. The direct product dimension pdim (G) (c.f. [10], [12]) is the minimum numbert such thatG can be embedded into a product oft copies of complete graphsKn.

In [10], Lovász, Nešetřil and Pultr determined the direct product dimension of matchings and paths and gave sharp bounds for the product dimension of cycles, all logarithmic in the number of vertices.

Here we prove that pdim (G)≤cdlogn for any graph with maximum degreed andn vertices and show that up to a factor of\(1/\left( {\log d + \log \log \frac{n}{{2d}}} \right)\) this bound is the best possible.

We also study set representations of graphs. LetG=(V,E) be a graph andp≥1 an integer. A familyF={A x ,xV} of (not necessarily distinct) sets is called ap-intersection representation ofG if |AxAy|≥p⇔{x,y}∈E for every pairx, y of distinct vertices ofG. Let θ G (G) be the minimum size of |UF| taken over all intersection representations ofG. We also study the parameter θ(G)=\(\mathop {\min }\limits_p \) p (G)).

It turns out that these parameters can be bounded in terms of maximum degree and linear density of a graphG or its complement\(\bar G\). While for example, θ1(G)=|E(G)| holds ifG contains no triangle, N. Alon proved that θ1(G)≤cΔ(\(\bar G\)) logn, where Δ(\(\bar G\)) denotes the maximum degree of\(\bar G\). We extend this by showing that Δ(\(\bar G\)) can be replaced by ϱ(\(\bar G\)), the linear density of\(\bar G\). We also show that this bound is close to best possible as there are graphs with\(\theta _1 (G) \geqslant c_2 \frac{{\Delta ^2 (\bar G)}}{{\log \Delta (G)}}\) logn.

For the parameter θ we conjecture that
$$\theta (G) \leqslant c\Delta (\bar G)^{1 + \varepsilon } \log n$$
for some constantc not dependent on Δ(\(\bar G\)) orn and show that θ(G)≤cΔ(\(\bar G\)) log2 Δ(\(\bar G\)) logn ifG is bipartite. This is, up to the factor 1/log2Δ(\(\bar G\)) best possible.

Finally, we give an upper bound on θ(G) in terms of Δ(G) and prove θ(G)≤cΔ2(G) logn.

Mathematics Subject Classification (1991)

05 C 

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References

  1. [1]
    N. Alon: Covering graphs by the minimum number of equivalence relations.Combinatorica,6 (3) (1986), 201–206.Google Scholar
  2. [2]
    M. S. Chung, andD. B. West: Thep-intersection number of a complete bipartite graph and orthogonal double coverings of a clique, to appear.Google Scholar
  3. [3]
    N. Eaton, R. Gould, andV. Rödl: Onp-intersection representations, to appear.Google Scholar
  4. [4]
    P. Erdős, A.W. Goodman, andL. Pósa: The representation of a graph by set intersections,Canad. J. of Math. 18 (1966), 106–112.Google Scholar
  5. [5]
    P. Erdős andL. Lovász: Problems and results on 4-chromatic hypergraphs and some related questions, In:Colloquia Mathematica Societatis János Bolyai, 10. Infinite and Finite Sets, Vol. II., North-Holland Publishing, 1975.Google Scholar
  6. [6]
    Z. Füredi: A random representation of the complete bipartite graph, to appear.Google Scholar
  7. [7]
    Z. Füredi, andJ. Kahn: On the dimensions of ordered sets of bounded degree,Order,3 (1986), 15–20.Google Scholar
  8. [8]
    W. Hoeffding, Probability inequalities for sums of bounded random variables.J. of the Amer. Math. Stat. Assoc. 58 (1963), 13–30.Google Scholar
  9. [9]
    M. S. Jacobson, A. E. Kézdy, andD. B. West: The 2-intersection number of paths and bounded-degree trees, to appear.Google Scholar
  10. [10]
    L. Lovász, J. Nešetřil, andA. Pultr: On a product dimension of graphs,J. of Combin. Theory, Ser. B,29 (1980), 47–67.Google Scholar
  11. [11]
    T. Luczak, A. Ruciński, andB. Voigt: Ramsey properties of random graphs,J. of Combin. Theory, Ser. B,56 (1), (1992), 66.Google Scholar
  12. [12]
    J. Nešetřil, andV. Rödl: A simple proof of the Galvin-Ramsey Property of the class of all finite graphs and a dimension of a graph,Discrete Mathematics,23 (1978), 49–55.Google Scholar
  13. [13]
    S. Poljak, andV. Rödl: Orthogonal partitions and covering of graphs,Czechoslovak Mathematical Journal,30 (105), (1980), 475–485.Google Scholar
  14. [14]
    J. Reitman, V. Rödl, andE. Šiňajová: On embedding of graphs into Euclidean Spaces of small dimension,J. of Combin. Theory, Ser. B 54 (1992).Google Scholar

Copyright information

© Akadémiai Kiadó 1996

Authors and Affiliations

  • Nancy Eaton
    • 1
  • Vojtěch Rödl
    • 2
  1. 1.University of Rhode IslandKingston
  2. 2.Emory UniversityAtlantaUSA

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