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.
Research supported in part by the Sloan Foundation, Grant No. 93-6-6.
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References
N. Alon, Decomposition of the complete r-graph into complete r-partite r-graphs, Graphs and Combinatorics 2 (1986), 95–100.
N. Alon, R. A. Brualdi and B. L. Shader, Multicolored forests in bipartite decompositions of graphs, J. Combinatorial Theory, Ser. B (1991), 143–148.
N. G. de Bruijn and P. Erdős, On a combinatorial problem, Indagationes Math. 20 (1948), 421–423.
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.
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.
A. Felzenbaum and M. A. Perles, Private communication.
R. L. Graham and L. Lovász, Distance matrix polynomials of trees, Advances in Math. 29 (1978), 60–88.
R. L. Graham and H. O. Pollak, On the addressing problem for loop switching, Bell Syst. Tech. J. 50 (1971), 2495–2519.
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.
J. Kasem, Neighborly families of boxes, Ph. D. Thesis, Hebrew University, Jerusalem, 1985.
L. Lovász, Combinatorial Problems and Exercises, Problem 11.22, North Holland, Amsterdam 1979.
J. Pach and P. Agarwal, Combinatorial Geometry, DIM ACS Tech. Report 41–51, 1991 (to be published by J. Wiley).
G. W. Peck, A new proof of a theorem of Graham and Pollak, Discrete Math. 49 (1984), 327–328.
M. A. Perles, At most 2 d+1 neighborly simplices in E d, Annals of Discrete Math. 20 (1984), 253–254.
J. Zaks, Bounds on neighborly families of convex polytopes, Geometriae Dedicata 8 (1979), 279–296.
J. Zaks, Neighborly families of 2 d d-simplices in E d, Geometriae Dedicata 11 (1981), 505–507.
J. Zaks, Amer. Math. Monthly 92 (1985), 568–571.
H. Tverberg, On the decomposition of K n into complete bipartite graphs, J. Graph Theory 6 (1982), 493–494.
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© 1997 Springer-Verlag Berlin Heidelberg
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Alon, N. (1997). Neighborly Families of Boxes and Bipartite Coverings. In: Graham, R.L., Nešetřil, J. (eds) The Mathematics of Paul Erdös II. Algorithms and Combinatorics, vol 14. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-60406-5_3
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DOI: https://doi.org/10.1007/978-3-642-60406-5_3
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