Advertisement

Results in Configuration Ordering

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

Abstract

In this chapter, we look at configuration orderings for block designs. We discuss standard configuration ordering, a listing of the blocks of a design such that sets of consecutive blocks are isomorphic to a given configuration. We present all the known results within the configuration ordering framework. The majority of results in this area arise from considering the Hamiltonicity of the block-intersection graph. Configuration orderings imply the existence of configuration decompositions. We then turn to generalized configuration orderings where a set of allowable configurations is permitted and similarly review all known results. We conclude the chapter with a look at configuration orderings for graph decompositions other than those represented by balanced incomplete block designs and pairwise balanced designs. This chapter will be of particular interest to the researcher interested in seeing all known results collected within a unified framework. This chapter will also be of interest to the researcher wanting to see the relatively recent work in generalized configuration ordering. Finally, this chapter will be of interest to the researcher wanting to see orderings for designs other than balanced incomplete block designs and pairwise balanced designs.

References

  1. 1.
    Adachi, T.: Optimal ordering for the complete tripartite graph K 9, 9, 9. In: Nonlinear Analysis and Convex Analysis, edited by W. Takahashi and T. Tanaka, pp. 1–10. Yokohama Publ., Yokohama (2007)Google Scholar
  2. 2.
    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 Mathematics, vol. 111, pp. 1–4. American Mathematical Society, Providence, RI (1990)Google Scholar
  3. 3.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Buck, M., Wiedemann, D.: Gray codes with restricted density. Discrete Math. 48, 163–171 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Case, G.A., Pike, D.A.: Pancyclic PBD block-intersection graphs. Discrete Math. 308, 896–900 (2008)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Chase, P.J.: Combination generation and Graylex ordering. Congr. Numer. 69, 215–242 (1989)MathSciNetGoogle Scholar
  7. 7.
    Cohen, M.B., Colbourn, C.J.: Optimal and pessimal orderings of Steiner triple systems in disk arrays. Theoret. Comput. Sci. 297, 103–117 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Cohen, M.B., Colbourn, C.J.: Ladder orderings of pairs and RAID performance. Disc. Appl. Math. 138(1), 35–46 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Cohen, M.B., Colbourn, C.J., Fronček, D.: Cluttered orderings for the complete graph. In: Lecture Notes in Computer Science, vol. 2108, pp. 420–431 (2001)CrossRefGoogle Scholar
  10. 10.
    Colbourn, C.J., Dinitz, J.H. (eds.): Handbook of Combinatorial Designs, second edn. Chapman & Hall/CRC, Boca Raton, FL (2007)zbMATHGoogle Scholar
  11. 11.
    Colbourn, C.J., Rosa, A.: Triple Systems. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York (1999)zbMATHGoogle Scholar
  12. 12.
    Colbourn, M.J., Johnstone, J.K.: Twofold triple systems with a minimal change property. Ars Combin. 18, 151–160 (1984)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Dewar, M.: Gray codes, universal cycles and configuration orderings for block designs. Ph.D. thesis, Carleton University, Ottawa, ON (2007)Google Scholar
  14. 14.
    Dinitz, J.H.: Starters, chap. VI.55, pp. 622–628. In: Colbourn and Dinitz [10] (2007)Google Scholar
  15. 15.
    Dinitz, J.H., Dukes, P., Stinson, D.R.: Sequentially perfect and uniform one-factorizations of the complete graph. Electron. J. Combin. 12, Research paper 1, (electronic, 12 pp.) (2005)Google Scholar
  16. 16.
    Eades, P., Hickey, M., Read, R.C.: Some Hamilton paths and a minimal change algorithm. J. Assoc. Comput. Mach. 31(1), 19–29 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Harary, F., Robinson, R.W., Wormald, N.C.: Isomorphic factorisations I: complete graphs. Trans. Amer. Math. Soc. 242, 243–260 (1978)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Hare, D.R.: Cycles in the block-intersection graph of pairwise balanced designs. Discrete Math. 137, 211–221 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Horák, P., Rosa, A.: Decomposing Steiner triple systems into small configurations. Ars Combin. 26, 91–105 (1988)Google Scholar
  21. 21.
    Horák, P., Rosa, A.: Private communication (2005)Google Scholar
  22. 22.
    Jesso, A.: The Hamiltonicity of block-intersection graphs. Master’s thesis, Memorial University of Newfoundland, St. John’s, NF (2010)Google Scholar
  23. 23.
    Jesso, A.T.: Private communication (2011)Google Scholar
  24. 24.
    Jesso, Andrew T.(3-NF); Pike, David A.(3-NF); Shalaby, Nabil(3-NF) Hamilton cycles in restricted block-intersection graphs. (English summary) Des. Codes Cryptogr. 61(3), 345–353 (2011)Google Scholar
  25. 25.
    Mamut, A., Pike, D.A., Raines, M.E.: Pancyclic BIBD block-intersection graphs. Discrete Math. 284, 205–208 (2004)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Momihara, K., Jimbo, M.: Some constructions for block sequences of Steiner quadruple systems with error-correcting consecutive unions. J. Combin. Des. 16(2), 152–163 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Momihara, K., Jimbo, M.: On a cyclic sequence of a packing by triples with error-correcting consecutive unions. Util. Math. 78, 93–105 (2009)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Müller, M., Adachi, T., Jimbo, M.: Cluttered orderings for the complete bipartite graph. Discrete Appl. Math. 152(1–3), 213–228 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Pike, D.A., Vandell, R.C., Walsh, M.: Hamiltonicity and restricted block-intersection graphs of t-designs. Discrete Math. 309, 6312–6315 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Rodney, P.: Balance in tournament designs. Ph.D. thesis, University of Toronto, Toronto, ON (1993)Google Scholar
  31. 31.
    Ruskey, F.: Adjacent interchange generation of combinations. J. Algorithms 9(2), 162–180 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Simmons, G.J., Davis, J.A.: Pair designs. Comm. Statist. 4, 255–272 (1975)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Stevens, B.: Maximally pair separated round robin tournaments: ordering the blocks of a design. Bull. Inst. Combin. Appl. 52, 21–32 (2008)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Varma, B.N.: On pancyclic line graphs. Congr. Numer. 54, 203–208 (1986)MathSciNetGoogle Scholar
  35. 35.
    Venkaiah, V.C., Ramanjaneyulu, K.: Sequentially perfect 1-factorization and cycle structure of patterned factorization of \({K}_{{2}^{n}}\) (2011). Presented at CanaDAM 2011 conference, Victoria, BC.Google Scholar
  36. 36.
    de Werra, D.: Some models of graphs for scheduling sports competitions. Discrete Appl. Math. 21(1), 47–65 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    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