Skip to main content
SpringerLink
Log in

The Spectrum of Tetrahedral Quadruple Systems

  • Original Paper
  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

An ordered analogue of quadruple systems is tetrahedral quadruple systems. A tetrahedral quadruple system of order v and index λ, TQS(v, λ), is a pair \({(S, \mathcal{T})}\) where S is a finite set of v elements and \({\mathcal{T}}\) is a family of oriented tetrahedrons of elements of S called blocks, such that every directed 3-cycle on S is contained in exactly λ blocks of \({\mathcal{T}}\) . When λ = 1, the spectrum problem of TQS(v, 1) has been completely determined. It is proved that a TQS(v, λ) exists if and only if λ(v − 1)(v − 2) ≡ 0 (mod 3), λv(v − 1)(v − 2) ≡ 0 (mod 4) and v ≥ 4.

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. Hananni H.: On quadruple systems. Can. J. Math. 12, 145–157 (1960)

    Article  Google Scholar 

  2. Hananni H.: On some tactical configurations. Can. J. Math. 15, 702–722 (1963)

    Article  Google Scholar 

  3. Stojaković Z., Madaras R.: On tetrahedral quadruple systems. Utilitas Math. 29, 19–26 (1986)

    MATH  MathSciNet  Google Scholar 

  4. Hartman A., Phelps K.T.: Steiner quadruple systems. In: Dinitz, J.H., Stinson, D.R. (eds) Contemporary Design theory, pp. 205–240. Wiley, New York (1992)

    Google Scholar 

  5. Hartman A., Phelps K.T.: The spectrum of tetrahedral quadruple systems. Utilitas Math. 37, 181–188 (1990)

    MATH  MathSciNet  Google Scholar 

  6. Ji L.: Purely tetrahedral quadruple systems. Sci. China Ser. A 49, 1327–1340 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  7. Ji L.: On the 3BD-closed set B3({4, 5, 6}). J. Combin. Des. 12, 92–102 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  8. Mills W.H.: On the existence of H designs. Congr. Numer. 79, 129–141 (1990)

    MATH  MathSciNet  Google Scholar 

  9. Ji L., Zhu L.: Constructions for Steiner quadruple systems with a spanning block design. Discrete Math. 261, 347–360 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  10. Ji L.: An improvement on H designs. J. Combin. Des. 17, 25–35 (2009)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Nantong Vocational College, Nantong, 226007, People’s Republic of China

    Jian Wang

  2. Department of Foundation, Suzhou Vocational University, Suzhou, 215104, People’s Republic of China

    Miao Liang

  3. Department of Mathematics, Suzhou University, Suzhou, 215006, People’s Republic of China

    Beiliang Du

Authors
  1. Jian Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Miao Liang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Beiliang Du
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Beiliang Du.

Additional information

This work was supported by the National Natural Science Foundation of China (Grant No. 10871143).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, J., Liang, M. & Du, B. The Spectrum of Tetrahedral Quadruple Systems. Graphs and Combinatorics 27, 593–602 (2011). https://doi.org/10.1007/s00373-010-0985-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-010-0985-y

Keywords

Search

Navigation