Neighborly Families of Boxes and Bipartite Coverings

  • Noga AlonEmail author


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 \(\Theta (k{n}^{1/k})\). 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.


  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.MathSciNetzbMATHCrossRefGoogle 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, 1985Google Scholar
  11. 11.
    L. Lovász, Combinatorial Problems and Exercises, Problem 11.22, North Holland, Amsterdam 1979.Google Scholar
  12. 12.
    J. Pach and P. K. Agarwal, Combinatorial Geometry, John Wiley and Sons, Inc., New York, 1995.zbMATHCrossRefGoogle 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 2d + 1 neighborly simplices inE 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 2d d-simplices inE 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 ofK n into complete bipartite graphs, J. Graph Theory 6 (1982), 493–494.MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Institute for Advanced StudyPrincetonUSA
  2. 2.Tel Aviv UniversityTel AvivIsrael

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