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Graphs and Combinatorics

, Volume 33, Issue 4, pp 845–858 | Cite as

An Unoriented Variation on de Bruijn Sequences

  • Christie S. Burris
  • Francis C. Motta
  • Patrick D. Shipman
Original Paper
  • 80 Downloads

Abstract

For positive integers kn, a de Bruijn sequence B(kn) is a finite sequence of elements drawn from k characters whose subwords of length n are exactly the \(k^n\) words of length n on k characters. This paper studies the unoriented de Bruijn sequence uB(kn), an analog to de Bruijn sequences, but for which the sequence is read both forwards and backwards to determine the set of subwords of length n. We show that nontrivial unoriented de Bruijn sequences of optimal length exist if and only if k is two or odd and n is less than or equal to 3. Unoriented de Bruijn sequences for any k, n may be constructed from certain Eulerian trails in Eulerizations of unoriented de Bruijn graphs.

Keywords

de Bruijn sequence de Bruijn graph Eulerian trail 

References

  1. 1.
    de Bruijn, N.G.: A combinatorial problem. Nederl. Akad. Wetensch. Proc. 49, 758–764 (1946)MathSciNetMATHGoogle Scholar
  2. 2.
    de Bruijn N. G.: Acknowledgement of priority to C. Flye Sainte-Marie on the counting of circular arrangements of \(2^n\) zeros and ones that show each n-letter word exactly once, Technological University Eindhoven Report 75-WSK-06 1-14 (1975)Google Scholar
  3. 3.
    Edmonds, J., Johnson, E.L.: Matching Euler tours and the Chinese postman problem. Math. Program. 5, 88–124 (1973)CrossRefMATHGoogle Scholar
  4. 4.
    Esfahanian, A.-H., Hakimi, S.L.: Fault-tolerant routing in de Bruijn communication networks. IEEE Trans. Comput. 34(9), 777–788 (1985)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Fleury, M.: Deux problmes de Géométrie de situation. J. de mathématiques élémentaires 2nd Ser 2, 257–261 (1883)Google Scholar
  6. 6.
    Hierholzer, C.: Über die Möglichkeit, einen Linienzug ohne Wiederholung und ohne Unterbrechung zu umfahren. Math. Ann. 6(1), 30–32 (1873)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Kuo, J., Fu, H.-L.: On the diameter of the generalized undirected de Bruijn graphs UGB(n, m), \(n^{2} < m \le n^3\). Networks 52(4), 180–182 (2008)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Kari, L., Xu, Z.: de Bruijn sequences revisited. In: Dömösi, P., Iván, Sz. (eds.) Automata and Formal Languages, Proceedings 2011, pp. 241–254 (2016)Google Scholar
  9. 9.
    Lu, C., Xu, J., Zhang, K.: On \((d,2)\)-dominating numbers of binary undirected de Bruijn graphs. Discret. Appl. Math. 105(13), 137–145 (2000)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Motta, F.C., Shipman, P.D., Springer, B.: A point of tangency between combinatorics and differential geometry. Am. Math. Mon. 122, 52–55 (2015)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Motta, F.C., Shipman, P.D., Springer, B.: Optimally topologically transitive orbits in discrete-time dynamical systems. Am. Math. Mon. 123(2), 115–135 (2016)CrossRefMATHGoogle Scholar

Copyright information

© Springer Japan 2017

Authors and Affiliations

  • Christie S. Burris
    • 1
  • Francis C. Motta
    • 2
  • Patrick D. Shipman
    • 1
  1. 1.Department of MathematicsColorado State UniversityFort CollinsUSA
  2. 2.Department of MathematicsDuke UniversityDurhamUSA

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