Background

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

Abstract

In this chapter we provide the basic background necessary for understanding subsequent chapters. We begin with an introduction to the graph theory, designs and design theory concepts used throughout this monograph. The second half of the chapter deals with more advanced concepts, including configurations, Gray codes and universal cycles (Ucycles). We survey a broad range of results in existence and construction of combinatorial Gray codes and Ucycles for permutations, partitions and k-subsets of n-sets. Proofs are included to illustrate techniques in this area, especially when they shed light on similar techniques used in proving results on ordering the blocks of designs.This chapter is recommended to the reader who wants to read subsequent chapters and feels they need a primer on the background topics: graph theory, design theory, and combinatorial orderings.

References

  1. 1.
    Beezer, R.A.: Counting configurations in designs. J. Combin. Theory Ser. A 96(2), 341–357 (2001)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bhat, G.S., Savage, C.D.: Balanced Gray codes. Electron. J. Combin. 3(1), Research Paper 25, (electronic, 11 pp.) (1996)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)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Brockman, G., Kay, B., Snively, E.E.: On universal cycles of labeled graphs. Electron. J. Combin. 17(1), Research Paper 4, (electronic, 9 pp.) (2010)Google Scholar
  5. 5.
    Brualdi, R.A.: Introductory Combinatorics. Prentice-Hall, Upper Saddle River, NJ (1999)MATHGoogle Scholar
  6. 6.
    Bryant, D., El-Zanati, S.: Graph Decompositions, chap. VI.24, pp. 477–486. In: Colbourn and Dinitz [14] (2007)Google Scholar
  7. 7.
    Buck, M., Wiedemann, D.: Gray codes with restricted density. Discrete Math. 48, 163–171 (1984)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Casteels, K.: Universal cycles for (n − 1)-partitions of an n-set. Master’s thesis, Carleton University, Ottawa, ON (2004)Google 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.
    Chvátal, V., Erdős, P.: A note on Hamiltonian circuits. Discrete Math. 2, 111–113 (1972)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Colbourn, C.J.: Combinatorial aspects of covering arrays. Matematiche (Catania) 59(1–2), 125–172 (2004)Google Scholar
  13. 13.
    Colbourn, C.J.: Covering Arrays, chap. VI.10, pp. 361–365. In: Colbourn and Dinitz [14] (2007)Google Scholar
  14. 14.
    Colbourn, C.J., Dinitz, J.H. (eds.): Handbook of Combinatorial Designs, second edn. Chapman & Hall/CRC, Boca Raton, FL (2007)MATHGoogle Scholar
  15. 15.
    Colbourn, C.J., Rosa, A.: Triple Systems. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York (1999)MATHGoogle Scholar
  16. 16.
    Curtis, D., Hines, T., Hurlbert, G., Moyer, T.: Near-universal cycles for subsets exist. SIAM J. Discrete Math. 23(3), 1441–1449 (2009)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Dewar, M.: Gray codes, universal cycles and configuration orderings for block designs. Ph.D. thesis, Carleton University, Ottawa, ON (2007)Google Scholar
  18. 18.
    Dewar, M., Proos, J., McInnes, L.: Monotone Gray codes for vectors of the form [ − m, m]k and [0, m]k. Submitted to Electron. J. Combin. (2012)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.
    Eades, P., McKay, B.: An algorithm for generating subsets of fixed size with a strong minimal change property. Inform. Process. Lett. 19(3), 131–133 (1984)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Fredricksen, H.: A survey of full length nonlinear shift register cycle algorithms. SIAM Rev. 24(2), 195–221 (1982)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Gordon, D.M., Stinson, D.R.: Coverings, chap. VI.11, pp. 365–373. In: Colbourn and Dinitz [14] (2007)Google Scholar
  23. 23.
    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
  24. 24.
    Grannell, M.J., Griggs, T.S., Mendelsohn, E.: A small basis for four-line configurations in Steiner triple systems. J. Combin. Des. 3(1), 51–59 (1995)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Grannell, M.J., Griggs, T.S., Whitehead, C.A.: The resolution of the anti-Pasch conjecture. J. Combin. Des. 8(4), 300–309 (2000)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Hartman, A.: Software and hardware testing using combinatorial covering suites. In: Graph Theory, Combinatorics and Algorithms, pp. 237–266. Springer, New York (2005)Google Scholar
  27. 27.
    Holroyd, A., Ruskey, F., Williams, A.: Shorthand universal cycles for permutations. Algorithmica, 64, 215–245 (2012)MathSciNetGoogle Scholar
  28. 28.
    Horák, P., Rosa, A.: Decomposing Steiner triple systems into small configurations. Ars Combin. 26, 91–105 (1988)Google Scholar
  29. 29.
    Hurlbert, G.H.: Universal cycles: on beyond de Bruijn. Ph.D. thesis, Rutgers University, New Brunswick, NJ (1990)Google Scholar
  30. 30.
    Hurlbert, G.H.: On universal cycles for k-subsets of an n-set. SIAM J. Disc. Math. 7(4), 598–604 (1994)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Hurlbert, G.H.: Multicover Ucycles. Discrete Math. 137, 241–249 (1995)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Hurlbert, G.H., Isaak, G.: Equivalence class universal cycles for permutations. Discrete Math. 149, 123–129 (1996)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Hurlbert, G.H., Johnson, T., Zahl, J.: On universal cycles for multisets. Discrete Math. 309, 5321–5327 (2009)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Jackson, B.W.: Universal cycles of k-subsets and k-permutations. Discrete Math. 117, 141–150 (1993)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Johnson, J.R.: Universal cycles for permutations. Discrete Math. 309, 5264–5270 (2009)MATHGoogle Scholar
  36. 36.
    Knuth, D.E.: The Art of Computer Programming, vol. 4. Addison-Wesley, Upper Saddle River, NJ (2005)Google Scholar
  37. 37.
    Kompelmaher, V.L., Liskovec, V.A.: Successive generation of permutations by means of a transposition basis. Kibernetika (Kiev) (3), 17–21 (1975)Google Scholar
  38. 38.
    Kreher, D.L., Stinson, D.R.: Combinatorial Algorithms: generation, Enumeration and Search. CRC Press, Boca Raton, FL (1999)MATHGoogle Scholar
  39. 39.
    Martin, M.H.: A problem in arrangements. Bull. Amer. Math. Soc. 40, 859–864 (1934)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Ruskey, F.: Adjacent interchange generation of combinations. J. Algorithms 9(2), 162–180 (1988)MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    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
  42. 42.
    Ruskey, F., Sawada, J., Williams, A.: Fixed-density de Bruijn sequences. SIAM J. Disc. Math. 26(2), 605–617 (2012)MATHCrossRefGoogle Scholar
  43. 43.
    Ruskey, F., Williams, A.: An explicit universal cycle for the (n − 1)-permutations of an n-set. ACM Trans. Algorithms 6(3), Art. 45, (electronic, 12 pp.) (2010)Google Scholar
  44. 44.
    Savage, C.D.: A survey of combinatorial Gray codes. SIAM Rev. 39(4), 605–629 (1997)MathSciNetMATHCrossRefGoogle Scholar
  45. 45.
    Savage, C.D., Winkler, P.: Monotone Gray codes and the middle levels problem. J. Combin. Theory. Ser. A 70(2), 230–248 (1995)MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    Sawada, J., Stevens, B., Williams, A.: De Bruijn sequences for the binary strings with maximum density. In: Proceedings of the 5th international conference on WALCOM: Algorithms and computation, pp. 182–190. Springer-Verlag, Berlin, Heidelberg (2011)Google Scholar
  47. 47.
    Sawada, J., Williams, A.: A Gray code for fixed-density necklaces and Lyndon words in constant amortized time. To appear in Theoret. Comput. Sci. (2011)Google Scholar
  48. 48.
    Slater, P.J.: Generating all permutations by graphical transpositions. Ars Combin. 5, 219–225 (1978)MathSciNetMATHGoogle Scholar
  49. 49.
    Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences. http://www.research.att.com/~njas/sequences/ (2004). Accessed June 2004
  50. 50.
    Stevens, B., Buskell, P., Ecimovic, P., Ivanescu, C., Malik, A., Savu, A., Vassilev, T., Verrall, H., Yang, B., Zhao, Z.: Solution of an outstanding conjecture: the non-existence of universal cycles with \(k = n - 2\). Discrete Math. 258, 193–204 (2002)MathSciNetMATHCrossRefGoogle Scholar
  51. 51.
    Stinson, D.R.: Combinatorial Designs: constructions and Analysis. Springer, New York (2004)MATHGoogle Scholar
  52. 52.
    Stinson, D.R., Wei, R., Yin, J.: Packings, chap. VI.40, pp. 550–556. In: Colbourn and Dinitz [14] (2007)Google Scholar
  53. 53.
    Vickers, V.E., Silverman, J.: A technique for generating specialized Gray codes. IEEE Trans. Comput. 29(4), 329–331 (1980)MathSciNetMATHCrossRefGoogle Scholar
  54. 54.
    Wagner, D.G., West, J.: Construction of uniform Gray codes. Congr. Numer. 80, 217–223 (1991)MathSciNetGoogle Scholar
  55. 55.
    Wallis, W.D.: Combinatorial Designs. Marcel Dekker, Inc., New York (1988)MATHGoogle Scholar
  56. 56.
    West, D.B.: Introduction to Graph Theory, second edn. Prentice Hall, Upper Saddle River, NJ (2001)Google Scholar
  57. 57.
    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
  58. 58.
    Williams, A.: Shift Gray codes. Ph.D. thesis, University of Victoria, Victoria, BC (2009)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

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