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Results in Configuration Ordering

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Ordering Block Designs

Part of the book series: CMS Books in Mathematics ((CMSBM))

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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.

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References

  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. 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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  4. Buck, M., Wiedemann, D.: Gray codes with restricted density. Discrete Math. 48, 163–171 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  5. Case, G.A., Pike, D.A.: Pancyclic PBD block-intersection graphs. Discrete Math. 308, 896–900 (2008)

    MathSciNet  MATH  Google Scholar 

  6. Chase, P.J.: Combination generation and Graylex ordering. Congr. Numer. 69, 215–242 (1989)

    MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cohen, M.B., Colbourn, C.J.: Ladder orderings of pairs and RAID performance. Disc. Appl. Math. 138(1), 35–46 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  10. Colbourn, C.J., Dinitz, J.H. (eds.): Handbook of Combinatorial Designs, second edn. Chapman & Hall/CRC, Boca Raton, FL (2007)

    MATH  Google Scholar 

  11. Colbourn, C.J., Rosa, A.: Triple Systems. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York (1999)

    MATH  Google Scholar 

  12. Colbourn, M.J., Johnstone, J.K.: Twofold triple systems with a minimal change property. Ars Combin. 18, 151–160 (1984)

    MathSciNet  MATH  Google Scholar 

  13. Dewar, M.: Gray codes, universal cycles and configuration orderings for block designs. Ph.D. thesis, Carleton University, Ottawa, ON (2007)

    Google Scholar 

  14. Dinitz, J.H.: Starters, chap. VI.55, pp. 622–628. In: Colbourn and Dinitz [10] (2007)

    Google Scholar 

  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. Eades, P., Hickey, M., Read, R.C.: Some Hamilton paths and a minimal change algorithm. J. Assoc. Comput. Mach. 31(1), 19–29 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  17. Harary, F., Robinson, R.W., Wormald, N.C.: Isomorphic factorisations I: complete graphs. Trans. Amer. Math. Soc. 242, 243–260 (1978)

    MathSciNet  MATH  Google Scholar 

  18. Hare, D.R.: Cycles in the block-intersection graph of pairwise balanced designs. Discrete Math. 137, 211–221 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  20. Horák, P., Rosa, A.: Decomposing Steiner triple systems into small configurations. Ars Combin. 26, 91–105 (1988)

    Google Scholar 

  21. Horák, P., Rosa, A.: Private communication (2005)

    Google Scholar 

  22. Jesso, A.: The Hamiltonicity of block-intersection graphs. Master’s thesis, Memorial University of Newfoundland, St. John’s, NF (2010)

    Google Scholar 

  23. Jesso, A.T.: Private communication (2011)

    Google Scholar 

  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. Mamut, A., Pike, D.A., Raines, M.E.: Pancyclic BIBD block-intersection graphs. Discrete Math. 284, 205–208 (2004)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  28. Müller, M., Adachi, T., Jimbo, M.: Cluttered orderings for the complete bipartite graph. Discrete Appl. Math. 152(1–3), 213–228 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  29. Pike, D.A., Vandell, R.C., Walsh, M.: Hamiltonicity and restricted block-intersection graphs of t-designs. Discrete Math. 309, 6312–6315 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  30. Rodney, P.: Balance in tournament designs. Ph.D. thesis, University of Toronto, Toronto, ON (1993)

    Google Scholar 

  31. Ruskey, F.: Adjacent interchange generation of combinations. J. Algorithms 9(2), 162–180 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  32. Simmons, G.J., Davis, J.A.: Pair designs. Comm. Statist. 4, 255–272 (1975)

    MathSciNet  MATH  Google Scholar 

  33. Stevens, B.: Maximally pair separated round robin tournaments: ordering the blocks of a design. Bull. Inst. Combin. Appl. 52, 21–32 (2008)

    MathSciNet  MATH  Google Scholar 

  34. Varma, B.N.: On pancyclic line graphs. Congr. Numer. 54, 203–208 (1986)

    MathSciNet  Google Scholar 

  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. de Werra, D.: Some models of graphs for scheduling sports competitions. Discrete Appl. Math. 21(1), 47–65 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  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 

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Authors and Affiliations

  1. School of Mathematics & Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario, Canada

    Megan Dewar & Brett Stevens

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  1. Megan Dewar
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  2. Brett Stevens
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Dewar, M., Stevens, B. (2012). Results in Configuration Ordering. In: Ordering Block Designs. CMS Books in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4325-4_4

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