Associativity and Non-Associativity of Some Hypergraph Products | Mathematics in Computer Science Skip to main content
Springer Nature Link
Log in

Associativity and Non-Associativity of Some Hypergraph Products

  • Published:
Mathematics in Computer Science Aims and scope Submit manuscript

Abstract

Several variants of hypergraph products have been introduced as generalizations of the strong and direct products of graphs. Here we show that only some of them are associative. In addition to the Cartesian product, these are the minimal rank preserving direct product, and the normal product. Counter-examples are given for the strong product as well as the non-rank-preserving and the maximal rank preserving direct product.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Japan)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Berge, C.: Hypergraphs: Combinatorics of finite sets, vol. 45. North-Holland, Amsterdam (1989)

    MATH  Google Scholar 

  2. Black, T.: Monotone properties of k-uniform hypergraphs are weakly evasive. In: Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science. ITCS ’15, pp. 383–391, New York, NY, USA (2015). ACM

  3. Bretto, A., Silvestre, Y., Vallée. T.: Cartesian product of hypergraphs: properties and algorithms. In: 4th Athens Colloquium on Algorithms and Complexity (ACAC 2009). vol. 4 of EPTCS. pp. 22–28 (2009)

  4. Dörfler, W.: Multiple Covers of Hypergraphs. Annals of the New York Academy of Sciences 319(1), 169–176 (1979)

    Article  MathSciNet  Google Scholar 

  5. Hammack, R., Imrich, W., Klavžar, S.: Handbook of Product Graphs. Discrete Mathematics and its Applications, 2nd edn. CRC Press (2011)

  6. Hellmuth, M., Lehner, F.: Fast factorization of cartesian products of (directed) hypergraphs. J. Theor. Comp. Sci. (2015) submitted

  7. Hellmuth, M., Ostermeier, L., Noll, M.: Strong products of hypergraphs: Unique prime factorization theorems and algorithms. Discr. Appl. Math. 171, 60–71 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hellmuth, M., Ostermeier, L., Stadler, P.F.: A survey on hypergraph products. Math. Comp. Sci. 6, 1–32 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Imrich, W.: Kartesisches Produkt von Mengensystemen und Graphen. Studia Sci. Math. Hungar. 2, 285–290 (1967)

    MathSciNet  MATH  Google Scholar 

  10. Imrich, W.: über das schwache Kartesische Produkt von Graphen. Journal of Combinatorial Theory 11(1), 1–16 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  11. Imrich, W., Izbicki, H.: Associative products of graphs. Monatshefte für Mathematik 80(4), 277–281 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kaveh, A., Alinejad, B.: Hypergraph products for structural mechanics. In: Topping, B.H.V. (ed), Proceedings of the Eleventh International Conference on Computational Structures Technology. Stirlingshire, UK (2012). Civil-Comp Press. Paper 266

  13. Kaveh, A., Alinejad, B.: Hypergraph products for structural mechanics. Advances in Engineering Software 80, 72–81 (2015). Civil-Comp

    Article  Google Scholar 

  14. Ostermeier, L., Hellmuth, M., Stadler, P.F.: The Cartesian product of hypergraphs. Journal of Graph Theory (2011)

  15. Sonntag, M.: Hamiltonian properties of the Cartesian sum of hypergraphs. J. Inf. Process. Cybern. 25(3), 87–100 (1989)

    MathSciNet  MATH  Google Scholar 

  16. Sonntag, M.: Hamiltonicity of the normal product of hypergraphs. J. Inf. Process. Cybern. 26(7), 415–433 (1990)

    MathSciNet  MATH  Google Scholar 

  17. Sonntag, M.: Hamiltonsche Eigenschaften von Produkten von Hypergraphen. Habilitation, Fakultät für Mathematik und Naturwissenschaften, Bergakademie Freiberg (1991)

  18. Zhu, X.: On the chromatic number of the product of hypergraphs. Ars Comb. 34, 25–31 (1992)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, VA, 23284-2014, USA

    Richard H. Hammack

  2. Department of Mathematics and Computer Science, University of Greifswald, Walther-Rathenau-Straße 47, 17487, Greifswald, Germany

    Marc Hellmuth

  3. Center for Bioinformatics, Saarland University, Building E 2.1, Room 413, P.O. Box 15 11 50, 66041, Saarbrücken, Germany

    Marc Hellmuth

  4. Bioinformatics Group, Department of Computer Science and Interdisciplinary Center for Bioinformatics, University of Leipzig, Härtelstraße 16-18, 04107, Leipzig, Germany

    Lydia Ostermeier & Peter F. Stadler

  5. Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103, Leipzig, Germany

    Peter F. Stadler

  6. RNomics Group, Fraunhofer Institut für Zelltherapie und Immunologie, Deutscher Platz 5e, 04103, Leipzig, Germany

    Peter F. Stadler

  7. Department of Theoretical Chemistry, University of Vienna, Währingerstraße 17, 1090, Vienna, Austria

    Peter F. Stadler

  8. Santa Fe Institute, 1399 Hyde Park Rd., Santa Fe, NM, 87501, USA

    Peter F. Stadler

Authors
  1. Richard H. Hammack
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marc Hellmuth
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Lydia Ostermeier
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Peter F. Stadler
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marc Hellmuth.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hammack, R.H., Hellmuth, M., Ostermeier, L. et al. Associativity and Non-Associativity of Some Hypergraph Products. Math.Comput.Sci. 10, 403–408 (2016). https://doi.org/10.1007/s11786-016-0276-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11786-016-0276-y

Keywords

Mathematics Subject Classification

Search

Navigation