List of derivatives and integrals in alternative calculi: Difference between revisions

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[[File:Complete-edge-coloring.svg|thumb|A partition of a [[complete graph]] into [[perfect matching]]s, the case ''r'' = 2 of Baranyai's theorem]]
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In [[combinatorics|combinatorial]] mathematics, '''Baranyai's theorem''' (proved by and named after [[Zsolt Baranyai]]) deals with the [[Glossary of graph theory#Independence|decompositions]] of complete [[hypergraph]]s.
 
==Statement of the theorem==
The statement of the result is that if <math>2\le r<k</math>  are natural numbers and ''r'' divides ''k'', then the complete [[hypergraph]] <math>K^k_r</math> decomposes into [[graph factorization|1-factors]]. <math>K^k_r</math> is a hypergraph with ''k'' vertices, in which every subset of ''r'' vertices forms a hyperedge; a 1-factor of this hypergraph is a set of hyperedges that touches each vertex exactly once, or equivalently a [[Partition of a set|partition]] of the vertices into subsets of size&nbsp;''r''. Thus, the theorem states that the ''k'' vertices of the hypergraph may be partitioned into subsets of ''r'' vertices in <math>\binom{k}{r}\frac{r}{k}</math> different ways, in such a way that each ''r''-element subset appears in exactly one of the partitions.
 
==History==
The ''r''&nbsp;=&nbsp;2 case can be rephrased as stating that every [[complete graph]] with an even number of vertices has an [[edge coloring]] whose number of colors equals its [[degree (graph theory)|degree]], or equivalently that its edges may be partitioned into [[perfect matching]]s. It may be used to schedule [[round-robin tournament]]s, and its solution was already known in the 19th century. The case that ''k''&nbsp;=&nbsp;2''r'' is also easy.
 
The ''r''&nbsp;=&nbsp;3 case was established by R. Peltesohn in 1936. The general case was proved by [[Zsolt Baranyai]] in 1975.
 
==References==
*{{citation|first=Zs.|last=Baranyai|authorlink=Zsolt Baranyai|contribution=On the factorization of the complete uniform hypergraph|title=Infinite and Finite Sets, Proc. Coll. Keszthely, 1973|editor1-first=A.|editor1-last=Hajnal|editor1-link=András Hajnal|editor2-first=R.|editor2-last=Rado|editor2-link=Richard Rado|editor3-first=V. T.|editor3-last=Sós|editor3-link=Vera T. Sós|series=Colloquia Math. Soc. János Bolyai|volume=10|publisher=North-Holland|year=1975|pages=91–107}}.
*{{citation|author1-link=Jack van Lint|first1=J. H.|last1=van Lint|author2-link=R. M. Wilson|first2=R. M.|last2=Wilson|title=A Course in Combinatorics|edition=2nd|publisher=[[Cambridge University Press]]|year=2001}}.  
*{{citation|first=R.|last=Peltesohn|title=Das Turnierproblem für Spiele zu je dreien|series=[[Inaugural dissertation]]|location=Berlin|year=1936}}.
[[Category:Hypergraphs]]
[[Category:Theorems in combinatorics]]

Latest revision as of 21:19, 6 May 2014

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