Skip to main content
SpringerLink
Log in

Further Results on Existentially Closed Graphs Arising from Block Designs

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

Abstract

A graph is n-existentially closed (n-e.c.) if for any disjoint subsets A, B of vertices with \(\left| {A \cup B} \right| =n\), there is a vertex \(z \notin A \cup B\) adjacent to every vertex of A and no vertex of B. For a block design with block set \({\mathcal {B}}\), its block intersection graph is the graph whose vertex set is \({\mathcal {B}}\) and two vertices (blocks) are adjacent if they have non-empty intersection. In this paper, we investigate the block intersection graphs of pairwise balanced designs, and propose a sufficient condition for such graphs to be 2-e.c. In particular, we study the \(\lambda \)-fold triple systems with \(\lambda \ge 2\) and determine for which parameters their block intersection graphs are 1- or 2-e.c. Moreover, for Steiner quadruple systems, the block intersection graphs and their analogue called \(\{1\}\)-block intersection graphs are investigated, and the necessary and sufficient conditions for such graphs to be 2-e.c. are established.

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. Ananchuen, W., Caccetta, L.: On the adjacency properties of Paley graphs. Networks 23(4), 227–236 (1993)

    Article  MathSciNet  Google Scholar 

  2. Baker, C.A., Bonato, A., Brown, J.M.N.: Graphs with the \(3\)-e.c. adjacency property constructed from affine planes. J. Combin. Math. Combin. Comput. 46, 65–83 (2003)

    MathSciNet  MATH  Google Scholar 

  3. Baker, C.A., Bonato, A., Brown, J.M.N., Szőnyi, T.: Graphs with the \(n\)-e.c. adjacency property constructed from affine planes. Discrete Math. 308(5–6), 901–912 (2008)

    Article  MathSciNet  Google Scholar 

  4. Beth, T., Jungnickel, D., Lenz, H.: Design Theory Vol. 1, Encyclopedia Math. Appl., vol. 69, 2nd edn. Cambridge University Press, Cambridge (1999)

  5. Blass, A., Exoo, G., Harary, F.: Paley graphs satisfy all first-order adjacency axioms. J. Graph Theory 5(4), 435–439 (1981)

    Article  MathSciNet  Google Scholar 

  6. Bollobás, B., Thomason, A.: Graphs which contain all small graphs. Eur. J. Combin. 2(1), 13–15 (1981)

    Article  MathSciNet  Google Scholar 

  7. Bonato, A.: The search for \(n\)-e.c. graphs. Contrib. Discrete Math. 4(1), 40–53 (2009)

    MathSciNet  MATH  Google Scholar 

  8. Bonato, A., Cameron, K.: On an adjacency property of almost all graphs. Discrete Math. 231(1–3), 103–119 (2001)

    Article  MathSciNet  Google Scholar 

  9. Bonato, A., Holzmann, W.H., Kharaghani, H.: Hadamard matrices and strongly regular graphs with the \(3\)-e.c. adjacency property. Electron. J. Combin. 8(1), R1 (9pp) (2001)

    MathSciNet  MATH  Google Scholar 

  10. Cameron, P.J.: The random graph. In: The Mathematics of Paul Erdős II, Algorithms Combin., vol. 14, pp. 333–351. Springer, New York (1997)

    Google Scholar 

  11. Colbourn, C.J., Dinitz, J.H. (eds.): Handbook of Combinatorial Designs, 2nd edn. CRC Press, Boca Raton (2007)

    MATH  Google Scholar 

  12. Colbourn, C.J., Forbes, A.D., Grannell, M.J., Griggs, T.S., Kaski, P., Östergård, P.R., Pike, D.A., Pottonen, O.: Properties of the Steiner triple systems of order 19. Electron. J. Combin. 17(1), #R98 (2010)

    MathSciNet  MATH  Google Scholar 

  13. Colbourn, C.J., Rosa, A.: Triple Systems. Oxford University Press, Oxford (1999)

    MATH  Google Scholar 

  14. Dehon, M.: On the existence of \(2\)-designs \({S}_\lambda (2, 3, v)\) without repeated blocks. Discrete Math. 43(2–3), 155–171 (1983)

    Article  MathSciNet  Google Scholar 

  15. Erdős, P., Rényi, A.: Asymmetric graphs. Acta Math. Acad. Sci. Hungar. 14(3–4), 295–315 (1963)

    Article  MathSciNet  Google Scholar 

  16. Forbes, A., Grannell, M.J., Griggs, T.S.: Steiner triple systems and existentially closed graphs. Electron. J. Combin. 12(3), #R42 (2005)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  18. Hartman, A., Phelps, K.T.: Steiner quadruple systems. In: J. Dinitz, D. Stinson (eds.) Contemporary Design Theory: A Collection of Surveys, chap. 6, pp. 205–240. Wiley, New York (1992)

  19. Horsley, D., Pike, D.A., Sanaei, A.: Existential closure of block intersection graphs of infinite designs having infinite block size. J. Combin. Des. 19(4), 317–327 (2011)

    Article  MathSciNet  Google Scholar 

  20. Kaski, P., Östergård, P.R.: Classification Algorithms for Codes and Designs, Algorithms Comput. Math., vol. 15. Springer, New York (2006)

  21. Kaski, P., Östergård, P.R.: Extra material for “Classification Algorithms for Codes and Designs”. http://extras.springer.com/2006/978-3-540-28990-6/ (2006). Accessed 25 Feb 2019

  22. Kikyo, H., Sawa, M.: Köhler theory for countable quadruple systems. Tsukuba J. Math. 41(2), 189–213 (2017)

    Article  MathSciNet  Google Scholar 

  23. Köhler, E.: Allgemeine Schnittzahlen in \(t\)-designs. Discrete Math. 73(1–2), 133–142 (1989)

    MathSciNet  MATH  Google Scholar 

  24. Lindner, C.C., Rosa, A.: Steiner quadruple systems—a survey. Discrete Math. 22(2), 147–181 (1978)

    Article  MathSciNet  Google Scholar 

  25. Mathon, R., Rosa, A.: Tables of parameters of BIBDs with \(r\le 41\) including existence, enumeration, and resolvability results. Ann. Discrete Math 26, 275–308 (1985)

    MATH  Google Scholar 

  26. McKay, N.A., Pike, D.A.: Existentially closed BIBD block-intersection graphs. Electron. J. Combin. 14(4), #R70 (2007)

    MathSciNet  MATH  Google Scholar 

  27. Mendelsohn, N.S.: Intersection numbers of \(t\)-designs. In: L. Mirsky (ed.) Studies in Pure Mathematics, pp. 145–150. Academic Press, London (1971)

  28. Mendelsohn, N.S., Hung, S.H.Y.: On the Steiner systems \(S(3,4,14)\) and \(S(4,5,15)\). Utilitas Math. 1, 5–95 (1972)

    MathSciNet  Google Scholar 

  29. Pike, D.A., Sanaei, A.: Existential closure of block intersection graphs of infinite designs having finite block size and index. J. Combin. Des. 19(2), 85–94 (2011)

    Article  MathSciNet  Google Scholar 

  30. Piotrowski, W.: Notwendige Existenzbedingungen für Steinersysteme und ihre Verallgemeinerungen. Diplomarbeit Univ, Hamburg (1977)

    Google Scholar 

  31. Shen, H.: Embeddings of simple triple systems. Sci. China Ser. A 35(3), 283–291 (1992)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author is grateful to the anonymous referee for the valuable suggestions and comments.

Author information

Authors and Affiliations

  1. Department of Industrial Administration, Faculty of Science and Technology, Tokyo University of Science, 2641 Yamazaki, Noda-shi, Chiba, 278-8510, Japan

    Xiao-Nan Lu

Authors
  1. Xiao-Nan Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiao-Nan Lu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was supported in part by JSPS under Grant-in-Aid for Scientific Research (B) No. 15H03636 and No. 18H01133.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lu, XN. Further Results on Existentially Closed Graphs Arising from Block Designs. Graphs and Combinatorics 35, 1323–1335 (2019). https://doi.org/10.1007/s00373-019-02036-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-019-02036-z

Keywords

Mathematics Subject Classification

Search

Navigation