Skip to main content
SpringerLink
Log in

Combinatorial Trades: A Survey of Recent Results

  • Chapter
Designs 2002

Part of the book series: Mathematics and Its Applications ((MAIA,volume 563))

Abstract

The concept of a trade in a combinatorial structure has existed for some years now. However, in the last five years or so there has been a great deal of activity in the area. This survey paper builds upon the one by Khosrovshahi, Maimani and Torabi which appeared in Discrete Applied Mathematics (Volume 95, pp. 361–176) in 1999. In the short time since that survey appeared, the number of papers in the area has almost doubled. Trades are used in designs and latin squares; they also crop up in graph theory. In this paper the most recent work on trades is surveyed, with applications given.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 16.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. P. Adams, E.J. Billington, D.E. Bryant and A. Khodkar, The µ-way intersection problem for m-cycle systems, Discrete Mathematics 231 (2001), 27–56.

    Article  MATH  MathSciNet  Google Scholar 

  2. P. Adams, E.J. Billington, D.E. Bryant and E.S. Mahmoodian, The three-way intersection problem for latin squares, Discrete Math. (to appear).

    Google Scholar 

  3. P. Adams, E.J. Billington, D.E. Bryant and E.S. Mahmoodian, On the possible volumes of,µ-way latin trades, Aequationes Mathematicae (to appear).

    Google Scholar 

  4. P. Adams, A. Khodkar and C. Ramsay, Smallest defining sets of some STS (19), J. Combin. Math. Combin. Comput. 38 (2001), 225–230.

    MATH  MathSciNet  Google Scholar 

  5. R. Bean, D. Donovan, A. Khodkar and A.P. Street, Steiner trades that give rise to completely decomposable latin interchanges, Proc. Eleventh Australasian Workshop on Combinatorial Algroithms, University of Newcastle, NSW, 2000, pp. 17–30.

    Google Scholar 

  6. E.J. Billington, Further constructions of irreducible designs, Congressus Numerantium 35 (1982), 77–89.

    MathSciNet  Google Scholar 

  7. E.J. Billington, The intersection problem for combinatorial designs, Congressus Numerantium 92 (1993), 33–54.

    MathSciNet  Google Scholar 

  8. E.J. Billington, Decomposing complete tripartite graphs into cycles of lengths 3 and 4, Discrete Mathematics 197/198 (1999), 123–135.

    Google Scholar 

  9. E.J. Billington, M. Gionfriddo and C.C. Lindner, The intersection problem for K 4 —e designs, J. Statistical Planning and Inference 58 (1997), 5–27.

    Article  MATH  MathSciNet  Google Scholar 

  10. E.J. Billington and D.G. Hoffman, Trades and graphs, Graphs and Combinatorics 17 (2001), 39–54.

    Article  MATH  MathSciNet  Google Scholar 

  11. E.J. Billington and D.G. Hoffman, Trade spectra of complete partite graphs, Discrete Mathematics (to appear).

    Google Scholar 

  12. E.J. Billington and D.G. Hoffman, Decomposition of complete tripartite graphs into gregarious 4-cycles, Discrete Mathematics (to appear)

    Google Scholar 

  13. D.E. Bryant, On the volume of trades in triple systems, Australasian Journal of Combinatorics 15 (1997), 161–176.

    MATH  MathSciNet  Google Scholar 

  14. D.E. Bryant, M. Grannell, T. Griggs and B. Maenhaut, On the volume of 4-cycle trades, submitted.

    Google Scholar 

  15. D.E. Bryant and B.M. Maenhaut, Defining sets of G-designs, Australasian Journal of Combinatorics 17 (1998), 257–266.

    MATH  MathSciNet  Google Scholar 

  16. N.J. Cavenagh, Graph decompositions of complete tripartite graphs using trades, M.Sc. Thesis, University of Queensland, 1998.

    Google Scholar 

  17. N.J. Cavenagh, Decompositions of complete tripartite graphs into triangles with an edge attached, Utilitas Mathematica (to appear).

    Google Scholar 

  18. N.J. Cavenagh and E.J. Billington, On decomposing complete tripartite graphs into 5-cycles, Australasian Journal of Combinatorics 22 (2000), 41–62.

    MATH  MathSciNet  Google Scholar 

  19. C. Delaney, B.D. Gray, K. Gray, B.M. Maenhaut, M.J. Sharry and A.P. Street, Pointwise defining sets and trade cores, Australasian Journal of Combinatorics 16 (1997), 51–76.

    MATH  MathSciNet  Google Scholar 

  20. D. Donovan, A. Howse and P. Adams, A discussion of latin interchanges, J. Combin. Math. Combin. Comput. 23 (1997), 161–182.

    MATH  MathSciNet  Google Scholar 

  21. Z. Eslami and G.B. Khosrovshahi, A complete classification of 3(11,4,4) designs with nontrivial automorphism group, Journal of Combinatorial Designs 8 (2000), 419–425.

    Article  MATH  MathSciNet  Google Scholar 

  22. Z. Eslami and G.B. Khosrovshahi, Some new 6-(14,7,4) designs, J. Combinatorial Theory series A 93 (2001), 141–152.

    Article  MATH  MathSciNet  Google Scholar 

  23. Z. Eslami and B. Tayfeh-Rezaie, On 2-(v, 3) trades of minimum volume, Australasian Journal of Combinatorics 19 (1999), 239–251.

    MATH  MathSciNet  Google Scholar 

  24. H.-L. Fu, On the construction of certain type of latin squares with prescribed intersections, Ph.D. Thesis, Auburn University, 1980.

    Google Scholar 

  25. C.-M. Fu and H.-L. Fu, The intersection problem of Latin squares, Graphs, designs and combinatorial geometries (Catania, 1989), J. Combin. Inform. System Sci. 15 (1990), 89–95.

    MATH  MathSciNet  Google Scholar 

  26. B.D. Gray, The maximum number of trades of volume four in a symmetric design, Utilitas Mathematica 52 (1997), 193–203.

    MATH  MathSciNet  Google Scholar 

  27. B.D. Gray and C. Ramsay, On the spectrum of Steiner (v, k, t) trades II, Graphs and Combinatorics 15 (1999), 405–415.

    Article  MATH  MathSciNet  Google Scholar 

  28. B.D. Gray and C. Ramsay, On the spectrum of Steiner (v, k, t) trades I, J. Combin. Math. Combin. Comput. 34 (2000), 133–158.

    MATH  MathSciNet  Google Scholar 

  29. N. Hamilton and A. Khodkar, On minimum possible volumes of strong Steiner trades. Australasian Journal of Combinatorics 20 (1999), 197–203.

    MATH  MathSciNet  Google Scholar 

  30. A.S. Hedayat, The theory of trade-off for t-designs, Coding Theory and Design Theory, Part II, IMA Vol. Math. Appl., 21, 101–126, Springer, New York, 1990.

    Google Scholar 

  31. A. Hedayat and H.L. Hwang, Construction of BIB designs with various support sizes — with special emphasis for v = 8 and k = 4, J. Combinatorial Theory, series A 36 (1984), 163–173.

    Article  MathSciNet  Google Scholar 

  32. P.D. Jenkins, Combinatorial trades, University of Queensland Honours Thesis, 2001.

    Google Scholar 

  33. A.D. Keedwell, Critical sets for latin squares, graphs and block designs: a survey, Congessus Numerantium 113 (1996), 231–245.

    MATH  MathSciNet  Google Scholar 

  34. A. Khodkar, Smallest defining sets for the 36 non-isomorphic twofold triple systems of order nine, J. Combin. Math. Combin. Comput. 17 (1995), 209–215.

    MATH  MathSciNet  Google Scholar 

  35. A. Khodkar and D.G. Hoffman, On the non-existence of Steiner (y, k, 2) trades with certain volumes, Australasian Journal of Cornbinatorics 18 (1998), 303–311.

    MATH  MathSciNet  Google Scholar 

  36. G.B. Khosrovshahi and E.S. Mahmoodian, A linear algebraic algorithm for reducing the support size of t-designs and to generate a basis for trades, Communications in Statistics, Simulation and Computation 16 (1987), 1015–1038.

    Article  MATH  Google Scholar 

  37. G.B. Khosrovshahi and H.R. Maimani, On 2-(v, 3) Steiner trades of small volumes, Ars Combinatoria 52 (1999), 199–220.

    MATH  MathSciNet  Google Scholar 

  38. G.B. Khosrovshahi, H.R. Maimani and R. Torabi, On trades: an update, Discrete Applied Mathematics 95 (1999), 361–376.

    Article  MATH  MathSciNet  Google Scholar 

  39. G.B. Khosrovshahi, A. Nowzari-Dalini and R. Torabi, Trading signed designs and some new 4-(12,5,4) designs, Designs, Codes and Cryptography 11 (1997), 279–288.

    Article  MATH  MathSciNet  Google Scholar 

  40. G.B. Khosrovshahi and R. Torabi, Maximal trades, Ars Combinatoria 51 (1999), 211–223.

    MATH  MathSciNet  Google Scholar 

  41. G.B. Khosrovshahi and H. Yousefi-Azari, Octahedrals in Steiner quadruple systems, Utilitas Mathematica 51 (1997), 33–39.

    MATH  MathSciNet  Google Scholar 

  42. B.M. Maenhaut, On the volume of 5-cycle trades, Graphs and Cornbinatorics 17 (2001), 315–328.

    Article  MATH  MathSciNet  Google Scholar 

  43. E.S. Mahmoodian and M. Mirzakhani, Decomposition of complete tripartite graphs into 5-cycles, in Combinatorics Advances (eds. C.J. Colbourn and E.S. Mahmoodian), Kluwer Academic Publishers (1995), 235–241.

    Chapter  Google Scholar 

  44. S. Milici and G. Quattrocchi, On the intersection problem for three Steiner triple systems. Ars Combinatoria 24A (1987), 174–194.

    MathSciNet  Google Scholar 

  45. C. Ramsay, An algorithm for enumerating trades in designs, with an application to defining sets, J. Combin. Math. Combin. Comput. 24 (1997), 3–31.

    MATH  MathSciNet  Google Scholar 

  46. C. Ramsay, On the index of simple trades, Australasian Journal of Combinatorics 20 (1999), 207–221.

    MATH  MathSciNet  Google Scholar 

  47. N. Soltankhah, On directed trades, Australas. J. Combin. 11 (1995), 59–66.

    MATH  MathSciNet  Google Scholar 

  48. A.P. Street, A survey of rreducible balanced incomplete block designs, Ars Combinatoria 19A (1985), 43–60.

    MATH  MathSciNet  Google Scholar 

  49. A.P. Street, Trades and defining sets, in C.J. Clbourn and J.H. Dinitz, CRC Handbook of Combinatorial Designs, CRC Publishing Co., IV.46 (1996), 474–478.

    Google Scholar 

  50. H.S. White, F.N. Cole and L.D. Cummings, Complete classification of the triad systems on fifteen elements, Memoirs of the National Academy of Sciences 14 (1916), 1–189.

    Google Scholar 

  51. N. Wormald, private communication, 2001.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, The University of Queensland, Brisbane, Qld 4072, Australia

    Elizabeth J. Billington

Authors
  1. Elizabeth J. Billington
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Southern Illinois University, USA

    W. D. Wallis

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Kluwer Academic Publishers

About this chapter

Cite this chapter

Billington, E.J. (2003). Combinatorial Trades: A Survey of Recent Results. In: Wallis, W.D. (eds) Designs 2002. Mathematics and Its Applications, vol 563. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0245-2_3

Download citation

  • DOI: https://doi.org/10.1007/978-1-4613-0245-2_3

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-7958-4

  • Online ISBN: 978-1-4613-0245-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation