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On Partitioning and Packing Products with Rectangles

Published online by Cambridge University Press:  12 September 2008

Rudolf Ahlswede  and
Ning Cai
Rudolf Ahlswede
Affiliation:
Universität Bielefeld, Fakultät für Mathematik, Postfach 100131, 33501 Bielefeld, Germany
Ning Cai
Affiliation:
Universität Bielefeld, Fakultät für Mathematik, Postfach 100131, 33501 Bielefeld, Germany
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Abstract

In [1] we introduced and studied for product hypergraphs where ℋi = (i,ℰi), the minimal size π(ℋn) of a partition of into sets that are elements of . The main result was that

if the ℋis are graphs with all loops included. A key step in the proof concerns the special case of complete graphs. Here we show that (1) also holds when the ℋi are complete d-uniform hypergraphs with all loops included, subject to a condition on the sizes of the i. We also present an upper bound on packing numbers.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 3 , Issue 4 , December 1994 , pp. 429 - 434
Copyright
Copyright © Cambridge University Press 1994

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References

[1]Ahlswede, R. and Cai, N. (1993) On extremal set partitions in Cartesian product spaces. Combinatorics, Probability & Computing 2 211–220.CrossRefGoogle Scholar
[2]Ahlswede, R. and Cai, N. (1993) On poset partitions and hypergraph products, Preprint 93–008 of SFB 343, Diskrete Strukturen in der Mathematik, Bielefeld.Google Scholar
[3]Rota, G. C. (1964) On the foundations of combinatorial theory I. Möbius functions. Z. Wahrscheinlichkeitstheorie 2 340–368.CrossRefGoogle Scholar
[4]Stanley, R. P. (1986) Enumerative Combinatorics, Vol. i, Wadsworth & Brooks/Coll, Advanced Books & Software, Monterey, California.CrossRefGoogle Scholar