3-Abelian Cubes Are Avoidable on Binary Alphabets

  • Robert Mercaş
  • Aleksi Saarela
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7907)

Abstract

A k-abelian cube is a word uvw, where u, v, w have the same factors of length at most k with the same multiplicities. Previously it has been known that k-abelian cubes are avoidable over a binary alphabet for k ≥ 5. Here it is proved that this holds for k ≥ 3.

Keywords

combinatorics on words k-abelian equivalence repetition-freeness 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Blanchet-Sadri, F., Kim, J.I., Mercaş, R., Severa, W., Simmons, S., Xu, D.: Avoiding abelian squares in partial words. Journal of Combinatorial Theory. Series A 119(1), 257–270 (2012)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Blanchet-Sadri, F., Mercaş, R., Scott, G.: A generalization of Thue freeness for partial words. Theoretical Computer Science 410(8-10), 793–800 (2009)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Carpi, A.: On Dejean’s conjecture over large alphabets. Theoretical Computer Science 385(1-3), 137–151 (2007)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Currie, J., Rampersad, N.: A proof of Dejean’s conjecture. Mathematics of Computation 80, 1063–1070 (2011)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Dekking, F.M.: Strongly nonrepetitive sequences and progression-free sets. Journal of Combinatorial Theory. Series A 27(2), 181–185 (1979)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Erdős, P.: Some unsolved problems. Magyar Tudományos Akadémia Matematikai Kutató Intézete 6, 221–254 (1961)Google Scholar
  7. 7.
    Huova, M., Karhumäki, J.: On the unavoidability of k-abelian squares in pure morphic words. Journal of Integer Sequences 16(2) (2013)Google Scholar
  8. 8.
    Huova, M., Karhumäki, J., Saarela, A.: Problems in between words and abelian words: k-abelian avoidability. Theoretical Computer Science 454, 172–177 (2012)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Huova, M., Karhumäki, J., Saarela, A., Saari, K.: Local squares, periodicity and finite automata. In: Calude, C.S., Rozenberg, G., Salomaa, A. (eds.) Rainbow of Computer Science. LNCS, vol. 6570, pp. 90–101. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  10. 10.
    Karhumäki, J., Puzynina, S., Saarela, A.: Fine and Wilf’s theorem for k-abelian periods. In: Yen, H.-C., Ibarra, O.H. (eds.) DLT 2012. LNCS, vol. 7410, pp. 296–307. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  11. 11.
    Keränen, V.: Abelian squares are avoidable on 4 letters. In: Proceedings of the 19th International Colloquium on Automata, Languages and Programming, pp. 41–52 (1992)Google Scholar
  12. 12.
    Manea, F., Mercaş, R.: Freeness of partial words. Theoretical Computer Science 389(1-2), 265–277 (2007)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Mercas, R., Saarela, A.: 5-abelian cubes are avoidable on binary alphabets. In: Proceedings of the 14th Mons Days of Theoretical Computer Science (2012)Google Scholar
  14. 14.
    Rao, M.: Last cases of Dejean’s conjecture. Theoretical Computer Science 412(27), 3010–3018 (2011)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Thue, A.: Über unendliche Zeichenreihen. Norske Vid. Selsk. Skr. I, Mat. Nat. Kl. Christiania 7, 1–22 (1906); Reprinted in Selected Mathematical Papers of Axel Thue. Nagell, T. (ed.) Universitetsforlaget, Oslo, Norway, pp. 139–158 (1977)Google Scholar
  16. 16.
    Thue, A.: Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Norske Vid. Selsk. Skr. I, Mat. Nat. Kl. Christiania 1, 1–67 (1912); Reprinted in Selected Mathematical Papers of Axel Thue. Nagell, T. (ed.) Universitetsforlaget, Oslo, Norway, pp. 413–478 (1977)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Robert Mercaş
    • 1
  • Aleksi Saarela
    • 2
  1. 1.Fakultät für InformatikOtto-von-Guericke-Universität MagdeburgMagdeburgGermany
  2. 2.Department of Mathematics and StatisticsUniversity of TurkuTurkuFinland

Personalised recommendations