Skip to main content
SpringerLink
Log in

Hanani triple systems

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

Hanani triple systems onv≡1 (mod 6) elements are Steiner triple systems having (v−1)/2 pairwise disjoint almost parallel classes (sets of pairwise disjoint triples that spanv−1 elements), and the remaining triples form a partial parallel class. Hanani triple systems are one natural analogue of the Kirkman triple systems onv≡3 (mod 6) elements, which form the solution of the celebrated Kirkman schoolgirl problem. We prove that a Hanani triple system exists for allv≡1 (mod 6) except forv ∈ {7, 13}.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. Assaf, E. Mendelsohn and D. R. Stinson,On resolvable coverings of pairs by triples, Utilitas Math.32 (1987), 67–74.

    MATH  MathSciNet  Google Scholar 

  2. T. Beth, D. Jungnickel and H. Lenz,Design Theory, Bibl. Institut, Mannheim, 1984.

    Google Scholar 

  3. C. J. Colbourn,Computing the chromatic index of Steiner triple systems, The Computer J.25 (1982), 338–339.

    MathSciNet  Google Scholar 

  4. C. J. Colbourn and M. J. Colbourn,The chromatic index of cyclic Steiner 2-designs, Int. J. Math., Math. Sci.5 (1982), 823–825.

    Article  MATH  MathSciNet  Google Scholar 

  5. C. J. Colbourn and M. J. Colbourn,Greedy colourings of Steiner triple systems, Ann. Discrete Math.18 (1983), 201–207.

    MATH  MathSciNet  Google Scholar 

  6. J. H. Dinitz and D. R. Stinson,A survey of Room squares, inContemporary Design Theory (J. H. Dinitz and D. R. Stinson, eds.), Wiley, New York, 1992, pp. 137–204.

    Google Scholar 

  7. M. Hall Jr.,Combinatorial Theory, 2nd ed., J. Wiley & Sons, Chichester, 1986.

    MATH  Google Scholar 

  8. H. Hanani,On resolvable balanced incomplete block designs, J. Combin. Theory (A)17 (1974), 275–289.

    Article  MATH  MathSciNet  Google Scholar 

  9. T. P. Kirkman,On a problem in combinations, Cambridge and Dublin Math. J.2 (1847), 191–204.

    Google Scholar 

  10. A. Kotzig and A. Rosa,Nearly Kirkman systems, Proc. 5th S.-E. Conf. Combinat., Graph Theory & Comput., Congr. Numer. X, Winnipeg, 1974, pp. 607–614.

  11. R. C. Mullin and S. A. Vanstone,Steiner systems and Room squares, Ann. Discrete Math.7 (1980), 95–104.

    Article  MATH  MathSciNet  Google Scholar 

  12. N. Pippenger and J. Spencer,Asymptotic behavior of the chromatic index for hypergraphs, J. Combin. Theory (A)51 (1989), 24–42.

    Article  MATH  MathSciNet  Google Scholar 

  13. D. K. Ray-Chaudhuri and R. M. Wilson,Solution of Kirkman’s schoolgirl problem, Proc. Symp. Pure Math.19 (1971), 187–209.

    MathSciNet  Google Scholar 

  14. R. Rees and D. R. Stinson,On resolvable group-divisible designs with block size 3, Ars Combinat.23 (1987), 107–120.

    MATH  MathSciNet  Google Scholar 

  15. A. Rosa and C. J. Colbourn,Colorings of block designs, inContemporary Design Theory (J. H. Dinitz and D. R. Stinson, eds.), Wiley, New York, 1992, pp. 401–430.

    Google Scholar 

  16. D. R. Stinson,Frames for Kirkman triple systems, Discrete Math.65 (1987), 289–300.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Combinatorics and Optimization, University of Waterloo, N2L 3G1, Waterloo, Ontario, Canada

    S. A. Vanstone, P. J. Schellenberg & C. J. Colbourn

  2. Department of Computer Science and Engineering, University of Nebraska, 68588, Lincoln, NE, USA

    D. R. Stinson

  3. Department of Mathematics and Statistics, McMaster University, L8S 4K1, Hamilton, Ontario, Canada

    A. Rosa & J. E. Carter

  4. Department of Mathematics and Statistics, Memorial University of Newfoundland, A1C 5S7, St. John’s, Newfoundland, Canada

    R. Rees

  5. Department of Industrial Engineering, University of Toronto, M5S 1A4, Toronto, Ontario, Canada

    M. W. Carter

Authors
  1. S. A. Vanstone
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. R. Stinson
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. P. J. Schellenberg
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. A. Rosa
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. R. Rees
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. C. J. Colbourn
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. M. W. Carter
    View author publications

    You can also search for this author in PubMed Google Scholar

  8. J. E. Carter
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Vanstone, S.A., Stinson, D.R., Schellenberg, P.J. et al. Hanani triple systems. Israel J. Math. 83, 305–319 (1993). https://doi.org/10.1007/BF02784058

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02784058

Keywords

Search

Navigation