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Neighborly Families of Boxes and Bipartite Coverings

  • Noga Alon
Part of the Algorithms and Combinatorics book series (AC, volume 14)

Summary

A bipartite covering of order k of the complete graph K n on n vertices is a collection of complete bipartite graphs so that every edge of K n lies in at least 1 and at most k of them. It is shown that the minimum possible number of subgraphs in such a collection is Θ(kn 1/k ). This extends a result of Graham and Pollak, answers a question of Felzenbaum and Perles, and has some geometric consequences. The proofs combine combinatorial techniques with some simple linear algebraic tools.

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References

  1. 1.
    N. Alon, Decomposition of the complete r-graph into complete r-partite r-graphs, Graphs and Combinatorics 2 (1986), 95–100.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    N. Alon, R. A. Brualdi and B. L. Shader, Multicolored forests in bipartite decompositions of graphs, J. Combinatorial Theory, Ser. B (1991), 143–148.Google Scholar
  3. 3.
    N. G. de Bruijn and P. Erdős, On a combinatorial problem, Indagationes Math. 20 (1948), 421–423.Google Scholar
  4. 4.
    P. Erdős, On sequences of integers none of which divides the product of two others, and related problems, Mitteilungen des Forschungsinstituts für Mat. und Mech., Tomsk, 2 (1938), 74–82.Google Scholar
  5. 5.
    P. Erdős and G. Purdy, Some extremal problems in combinatorial geometry, in: Handbook of Combinatorics (R. L. Graham, M. Grötschel and L. Lovász eds.), North Holland, to appear.Google Scholar
  6. 6.
    A. Felzenbaum and M. A. Perles, Private communication.Google Scholar
  7. 7.
    R. L. Graham and L. Lovász, Distance matrix polynomials of trees, Advances in Math. 29 (1978), 60–88.zbMATHCrossRefGoogle Scholar
  8. 8.
    R. L. Graham and H. O. Pollak, On the addressing problem for loop switching, Bell Syst. Tech. J. 50 (1971), 2495–2519.MathSciNetzbMATHGoogle Scholar
  9. 9.
    R. L. Graham and H. O. Pollak, On embedding graphs in squashed cubes, In: Lecture Notes in Mathematics 303, pp 99–110, Springer Verlag, New York-Berlin Heidelberg, 1973.Google Scholar
  10. 10.
    J. Kasem, Neighborly families of boxes, Ph. D. Thesis, Hebrew University, Jerusalem, 1985.Google Scholar
  11. 11.
    L. Lovász, Combinatorial Problems and Exercises, Problem 11.22, North Holland, Amsterdam 1979.zbMATHGoogle Scholar
  12. 12.
    J. Pach and P. Agarwal, Combinatorial Geometry, DIM ACS Tech. Report 41–51, 1991 (to be published by J. Wiley).Google Scholar
  13. 13.
    G. W. Peck, A new proof of a theorem of Graham and Pollak, Discrete Math. 49 (1984), 327–328.MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    M. A. Perles, At most 2 d+1 neighborly simplices in E d, Annals of Discrete Math. 20 (1984), 253–254.MathSciNetGoogle Scholar
  15. 15.
    J. Zaks, Bounds on neighborly families of convex polytopes, Geometriae Dedicata 8 (1979), 279–296.MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    J. Zaks, Neighborly families of 2 d d-simplices in E d, Geometriae Dedicata 11 (1981), 505–507.MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    J. Zaks, Amer. Math. Monthly 92 (1985), 568–571.MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    H. Tverberg, On the decomposition of K n into complete bipartite graphs, J. Graph Theory 6 (1982), 493–494.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Noga Alon
    • 1
    • 2
  1. 1.Institute for Advanced StudyPrincetonUSA
  2. 2.Department of MathematicsTel Aviv UniversityTel AvivIsrael

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