Extending factorizations of complete uniform hypergraphs

  • M. A. Bahmanian
  • Mike Newman


We consider when a given r-factorization of the complete uniform hypergraph on m vertices K m h can be extended to an s-factorization of K n h . The case of r = s = 1 was first posed by Cameron in terms of parallelisms, and solved by Häggkvist and Hellgren. We extend these results, which themselves can be seen as extensions of Baranyai's Theorem. For r=s, we show that the “obvious” necessary conditions, together with the condition that gcd(m,n,h)=gcd(n,h) are sufficient. In particular this gives necessary and sufficient conditions for the case where r=s and h is prime. For r<s we show that the obvious necessary conditions, augmented by gcd(m,n,h)=gcd(n,h), n≥2m, and \(1 \leqslant \frac{s}{r} \leqslant \frac{m}{k}\left[ {1 - \left( {\begin{array}{*{20}{c}} {m - k} \\ h \end{array}} \right)/\left( {\begin{array}{*{20}{c}} m \\ h \end{array}} \right)} \right]\) are sufficient, where k=gcd(m,n,h). We conclude with a discussion of further necessary conditions and some open problems.

Mathematics Subject Classification (2000)

05C70 05C65 05C15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    L. D. Andersen and A. J. W. Hilton: Generalized Latin rectangles, II, Embedding, Discrete Math. 31 (1980), 235–260.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    A. Bahmanian and M. Newman: Embedding factorizations for 3-uniform hypergraphs II: r-factorizations into s-factorizations, Electron. J. Combin. 23 (2016), paper 2.42, 14.MathSciNetzbMATHGoogle Scholar
  3. [3]
    A. Bahmanian and C. Rodger: Embedding factorizations for 3-uniform hypergraphs, J. Graph Theory 73 (2013), 216–224.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    M. A. Bahmanian: Detachments of hypergraphs I: The Berge-Johnson problem, Combin. Probab. Comput. 21 (2012), 483–495.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Zs. Baranyai: On the factorization of the complete uniform hypergraph, in: Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), Vol. I, 91–108. Colloq. Math. Soc. János Bolyai, Vol. 10. North-Holland, Amsterdam, 1975.Google Scholar
  6. [6]
    Zs. Baranyai and A. E. Brouwer: Extension Of Colourings Of The Edges Of A Complete (uniform Hyper)graph, CWI Technical Report ZW 91/77, 1977.Google Scholar
  7. [7]
    P. J. Cameron: Parallelisms of complete designs, Cambridge University Press, Cambridge-New York-Melbourne, 1976. London Mathematical Society Lecture Note Series, No. 23.CrossRefzbMATHGoogle Scholar
  8. [8]
    R. Häggkvist and T. Hellgren: Extensions of edge-colourings in hypergraphs, I, in: Combinatorics, Paul Erdős is eighty, Vol. 1, Bolyai Soc. Math. Stud., 215–238. János Bolyai Math. Soc., Budapest, 1993.Google Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of OttawaOttawa, OntarioCanada
  2. 2.Department of Mathematics and StatisticsIllinois State UniversityNormalUSA

Personalised recommendations