Near-optimum Universal Graphs for Graphs with Bounded Degrees

Extended Abstract
  • Noga Alon
  • Michael Capalbo
  • Yoshiharu Kohayakawa
  • Vojtěch Rödl
  • Andrzej Ruciński
  • Endre Szemerédi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2129)


Let \( \mathcal{H} \) be a family of graphs. We say that G is \( \mathcal{H} \)-universal if, for each H\( \mathcal{H} \), the graph G contains a subgraph isomorphic to H. Let \( \mathcal{H} \) (k, n) denote the family of graphs on n vertices with maximum degree at most k. For each fixed k and each n sufficiently large, we explicitly construct an \( \mathcal{H} \) (k, n)-universal graph Γ(k, n) with O(n 2 - 2/k (log n)1+8/k ) edges. This is optimal up to a small polylogarithmic factor, as Ω(n 2-2/k ) is a lower bound for the number of edges in any such graph.

En route, we use the probabilistic method in a rather unusual way. After presenting a deterministic construction of the graph Γ(k, n) we prove, using a probabilistic argument, that Γ(k, n) is \( \mathcal{H} \)(k, n)-universal. So we use the probabilistic method to prove that an explicit construction satisfies certain properties, rather than showing the existence of a construction that satisfies these properties.


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  1. 1.
    N. Alon and V. Asodi, Sparse universal graphs, Journal of Computational and Applied Mathematics, to appear.Google Scholar
  2. 2.
    N. Alon, M. Capalbo, Y. Kohayakawa, V. Rödl, A. Ruciński, and E. Szemerédi, Universality and tolerance, Proceedings of the 41st IEEE Annual Symposium on FOCS, pp. 14–21, 2000.Google Scholar
  3. 3.
    L. Babai, F. R. K. Chung, P. Erdos, R. L. Graham, J. Spencer, On graphs which contain all sparse graphs, Ann. Discrete Math., 12 (1982), pp. 21–26.zbMATHMathSciNetGoogle Scholar
  4. 4.
    S. N. Bhatt, F. Chung, F. T. Leighton and A. Rosenberg, Universal graphs for bounded-degree trees and planar graphs, SI AM J. Disc. Math. 2 (1989), 145–155.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    S. N. Bhatt and E. Leiserson, How to assemble tree machines, Advances in Computing Research, F. Preparata, ed., 1984.Google Scholar
  6. 6.
    M. Capalbo, A small universal graph for bounded-degree planar graphs, SODA (1999), 150–154.Google Scholar
  7. 7.
    F. R. K. Chung and R. L. Graham, On graphs which contain all small trees, J. Combin. Theory Ser. B, 24 (1978) pp. 14–23.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    F. R. K. Chung and R. L. Graham, On universal graphs, Ann. New York Acad. Sci., 319 (1979) pp. 136–140.CrossRefMathSciNetGoogle Scholar
  9. 9.
    F. R. K. Chung and R. L. Graham, On universal graphs for spanning trees, Proc. London Math. Soc., 27 (1983) pp. 203–211.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    F. R. K. Chung, R. L. Graham, and N. Pippenger, On graphs which contain all small trees II, Proc. 1976 Hungarian Colloquium on Combinatorics, 1978, pp. 213–223.Google Scholar
  11. 11.
    M. Capalbo and S. R. Kosaraju, Small universal graphs, STOC (1999), 741–749.Google Scholar
  12. 12.
    F. R. K. Chung, A. L. Rosenberg, and L. Snyder, Perfect storage representations for families of data structures, SI AM J. Alg. Disc. Methods., 4 (1983), pp. 548–565.zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    J. Friedman and N. Pippenger, Expanding graphs contain all small trees, Combi-natorica, 7 (1987), pp. 71–76.zbMATHMathSciNetGoogle Scholar
  14. 14.
    A. Hajnal and E. Szemerédi, Proof of a conjecture of Erdos, in Combinatorial Theory and its Applications, Vol. II (P. Erdos, A. Rényi, and V. T. Sós, eds.), Colloq. Math Soc. J. Bolyai 4, North Holland, Amsterdam 1970, 601–623.Google Scholar
  15. 15.
    L. Lovász and M. D. Plummer, Matching Theory, North Holland, Amsterdam (1986).zbMATHGoogle Scholar
  16. 16.
    V. Rödl, A note on universal graphs, Ars Combin., 11 (1981), 225–229.MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Noga Alon
    • 1
  • Michael Capalbo
    • 2
  • Yoshiharu Kohayakawa
    • 3
  • Vojtěch Rödl
    • 4
  • Andrzej Ruciński
    • 5
  • Endre Szemerédi
    • 6
  1. 1.Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact SciencesTel Aviv UniversityTel AvivIsrael
  2. 2.Department of Mathematical SciencesThe Johns Hopkins UniversityBaltimore
  3. 3.Instituto de Matemática e EstatýsticaUniversidade de São PauloBrazil
  4. 4.Department of Mathematics and Computer ScienceEmory UniversityAtlanta
  5. 5.Faculty of Mathematics and Computer ScienceAdam Mickiewicz UniversityPoznańPoland
  6. 6.Department of Computer ScienceRutgers University

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