• Megan Dewar
  • Brett Stevens
Part of the CMS Books in Mathematics book series (CMSBM)


In this introductory chapter, we consider what it means to order the blocks of a design. Block orderings can be classified into two broad groups: local and global. The focus of this monograph is local orderings and we look at various definitions of these orderings, including Gray codes, universal cycles (Ucycles), and configuration orderings. Our aim in this chapter is to place block orderings in the context of other well-known combinatorial orderings, to survey the broad themes in block ordering research, and to give a brief summary of the range of results presented in subsequent chapters.This chapter will be of particular interest to the reader who wants a quick introduction to the subject area, its connections to other combinatorial problems and the main directions—past and current—of the research.


  1. 1.
    Alspach, B., Heinrich, K., Mohar, B.: A note on Hamilton cycles in block-intersection graphs. In: Finite Geometries and Combinatorial Designs (Lincoln, NE, 1987), Contemporary Math, vol. 111, pp. 1–4. American Mathematical Society, Providence, RI (1990)Google Scholar
  2. 2.
    Anderson, I., Finizio, N.J.: Whist Tournaments, chap. VI.64, pp. 663–668. In: Colbourn and Dinitz [12] (2007)Google Scholar
  3. 3.
    Batzoglou, S., Istrail, S.: Physical mapping with repeated probes: the hypergraph superstring problem. J. Discrete Algorithms (Oxf.) 1(1), 51–76 (2000)Google Scholar
  4. 4.
    Bennett, F.E., Mahmoodi, A.: Directed Designs, chap. VI.20, pp. 441–444. In: Colbourn and Dinitz [12] (2007)Google Scholar
  5. 5.
    Bitner, J.R., Ehrlich, G., Reingold, E.M.: Efficient generation of the binary reflected Gray code and its applications. Comm. ACM 19(9), 517–521 (1976)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Bryant, D., Rodger, C.: Cycle Decompositions, chap. VI.12, pp. 373–382. In: Colbourn and Dinitz [12] (2007)Google Scholar
  7. 7.
    Buck, M., Wiedemann, D.: Gray codes with restricted density. Discrete Math. 48, 163–171 (1984)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Case, G.A., Pike, D.A.: Pancyclic PBD block-intersection graphs. Discrete Math. 308, 896–900 (2008)MathSciNetMATHGoogle Scholar
  9. 9.
    Chase, P.J.: Combination generation and Graylex ordering. Congr. Numer. 69, 215–242 (1989)MathSciNetGoogle Scholar
  10. 10.
    Chung, F., Diaconis, P., Graham, R.: Universal cycles for combinatorial structures. Discrete Math. 110, 43–59 (1992)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Cohen, M.B., Colbourn, C.J.: Optimal and pessimal orderings of Steiner triple systems in disk arrays. Theoret. Comput. Sci. 297, 103–117 (2003)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Colbourn, C.J., Dinitz, J.H. (eds.): Handbook of Combinatorial Designs, second edn. Chapman & Hall/CRC, Boca Raton, FL (2007)MATHGoogle Scholar
  13. 13.
    Colbourn, C.J., Rosa, A.: Triple Systems. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York (1999)MATHGoogle Scholar
  14. 14.
    Colbourn, M.J., Johnstone, J.K.: Twofold triple systems with a minimal change property. Ars Combin. 18, 151–160 (1984)MathSciNetMATHGoogle Scholar
  15. 15.
    Cooke, M., Dewar, M., North, C., Stevens, B.: Beckett Gray codes. Submitted to J. Combin. Math. Combin. Comput. (2009)Google Scholar
  16. 16.
    Dewar, M.: Gray codes, universal cycles and configuration orderings for block designs. Ph.D. thesis, Carleton University, Ottawa (2007)Google Scholar
  17. 17.
    Dinitz, J.H.: Howel Designs, chap. VI.29, pp. 499–504. In: Colbourn and Dinitz [12] (2007)Google Scholar
  18. 18.
    Dinitz, J.H., Fronček, D., Lamken, E.R., Wallis, W.D.: Scheduling a Tournament, chap. VI.51, pp. 591–606. In: Colbourn and Dinitz [12] (2007)Google Scholar
  19. 19.
    Eades, P., Hickey, M., Read, R.C.: Some Hamilton paths and a minimal change algorithm. J. Assoc. Comput. Mach. 31(1), 19–29 (1984)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Grannell, M.J., Griggs, T.S.: Configurations in Steiner triple systems. In: Combinatorial Designs and their Applications (Milton Keynes, 1997), vol. 403, pp. 103–126. Chapman & Hall/CRC, Boca Raton, FL (1999)Google Scholar
  21. 21.
    Hanani, H.: Decomposition of hypergraphs into octahedra. In: Second International Conference on Combinatorial Mathematics (New York, 1978), Annals of the New York Academy of Sciences, vol. 319, pp. 260–264. New York Academy of Sciences, New York (1979)Google Scholar
  22. 22.
    Hare, D.R.: Cycles in the block-intersection graph of pairwise balanced designs. Discrete Math. 137, 211–221 (1995)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Hartman, A., Phelps, K.T.: The spectrum of tetrahedral quadruple systems. Utilitas Math. 37, 181–188 (1990)MathSciNetMATHGoogle Scholar
  24. 24.
    Horák, P., Pike, D.A., Raines, M.E.: Hamilton cycles in block-intersection graphs of triple systems. J. Combin. Des. 7(4), 243–246 (1999)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Horák, P., Rosa, A.: Decomposing Steiner triple systems into small configurations. Ars Combin. 26, 91–105 (1988)Google Scholar
  26. 26.
    Hurlbert, G.H.: Universal cycles: on beyond de Bruijn. Ph.D. thesis, Rutgers University, New Brunswick, NJ (1990)Google Scholar
  27. 27.
    Hurlbert, G.H.: On universal cycles for k-subsets of an n-set. SIAM J. Disc. Math. 7(4), 598–604 (1994)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Jackson, B.W.: Universal cycles of k-subsets and k-permutations. Discrete Math. 117, 141–150 (1993)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Jesso, A.: The Hamiltonicity of block-intersection graphs. Master’s thesis, Memorial University of Newfoundland, St. John’s, NF (2010)Google Scholar
  30. 30.
    Johnson, J.R.: Universal cycles for permutations. Discrete Math. 309, 5264–5270 (2009)MATHGoogle Scholar
  31. 31.
    Lamken, E.R.: Balanced Tournament Designs, chap. VI.3, pp. 333–336. In: Colbourn and Dinitz [12] (2007)Google Scholar
  32. 32.
    Linek, V., Stevens, B.: Octahedral designs. J. Combin. Des. 18(5), 319–327 (2010)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Mamut, A., Pike, D.A., Raines, M.E.: Pancyclic BIBD block-intersection graphs. Discrete Math. 284, 205–208 (2004)MathSciNetMATHGoogle Scholar
  34. 34.
    Martin, W.J.: (t, m, s)-Nets, chap. VI.59, pp. 639–643. In: Colbourn and Dinitz [12] (2007)Google Scholar
  35. 35.
    Mendelsohn, E.: Mendelsohn Designs, chap. VI.35, pp. 528–534. In: Colbourn and Dinitz [12] (2007)Google Scholar
  36. 36.
    Pike, D.A., Vandell, R.C., Walsh, M.: Hamiltonicity and restricted block-intersection graphs of t-designs. Discrete Math. 309, 6312–6315 (2009)MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Preece, D.A., Colbourn, C.J.: Youden Squares and Generalized Youden Designs, chap. VI.65, pp. 668–674. In: Colbourn and Dinitz [12] (2007)Google Scholar
  38. 38.
    Rodney, P.: Balance in tournament designs. Ph.D. thesis, University of Toronto, Toronto, ON (1993)Google Scholar
  39. 39.
    Ruskey, F.: Adjacent interchange generation of combinations. J. Algorithms 9(2), 162–180 (1988)MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Ruskey, F.: Simple combinatorial Gray codes constructed by reversing sublists. In: Proceedings of the Fourth International Symposium (ISAAC ’93) held in Hong Kong, Dec 15–17, 1993. Edited by K.W. Ng, P. Raghavan, N.V. Balasubramanian and F.Y.L. Chin. Lecture Notes in Computer Science, vol. 762, Springer-Verlag, Berlin, pp. xiv+542 (1993). ISBN: 3-540-57568-5Google Scholar
  41. 41.
    Savage, C.D.: A survey of combinatorial Gray codes. SIAM Rev. 39(4), 605–629 (1997)MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    Simmons, G.J., Davis, J.A.: Pair designs. Comm. Statist. 4, 255–272 (1975)MathSciNetMATHGoogle Scholar
  43. 43.
    Street, A.P., Street, D.J.: Combinatorics of Experimental Design. Oxford Science Publications. The Clarendon Press Oxford University Press, New York (1987)MATHGoogle Scholar
  44. 44.
    Wallis, W., Yucas, J.L., Zhang, G.H.: Single-change covering designs. Des. Codes Cryptogr. 3, 9–19 (1992)MathSciNetCrossRefGoogle Scholar
  45. 45.
    West, D.B.: Introduction to Graph Theory. Prentice Hall, Upper Saddle River, NJ (2001)Google Scholar
  46. 46.
    Wilf, H.S.: Combinatorial algorithms: an update. CBMS-NSF Regional Conference Series in Applied Mathematics, 55. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1989)Google Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Megan Dewar
    • 1
  • Brett Stevens
    • 1
  1. 1.School of Mathematics & StatisticsCarleton UniversityOttawaCanada

Personalised recommendations