Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-26T09:55:58.951Z Has data issue: false hasContentIssue false
  • English
  • Français

5-Abelian cubes are avoidable on binary alphabets∗∗∗

Published online by Cambridge University Press:  31 July 2014

Robert Mercaş  and
Aleksi Saarela
Robert Mercaş
Affiliation:
Christian-Albrechts-Universität zu Kiel, Institut für Informatik, 24098 Kiel, Germany. [email protected]
Aleksi Saarela
Affiliation:
Department of Mathematics and Statistics, University of Turku, 20014 Turku, Finland; [email protected]
Get access

Abstract

A k-abelian cube is a word uvw, where the factors u, v, and w are either pairwise equal, or have the same multiplicities for every one of their factors of length at most k. Previously it has been shown that k-abelian cubes are avoidable over a binary alphabet for k ≥ 8. Here it is proved that this holds for k ≥ 5.

Keywords

Combinatorics on wordsk-abelian equivalencerepetition-freenesscube-freeness
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 48 , Issue 4: Special issue in the honor of the 14th "Journées Montoises d'Informatique Théorique" , October 2014 , pp. 467 - 478
Copyright
© EDP Sciences 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Blanchet-Sadri, F., Kim, J.I., Mercaş, R., Severa, W., Simmons, S. and Xu, D., Avoiding abelian squares in partial words. J. Combin. Theory Ser. A 119 (2012) 257–270. Google Scholar
Blanchet-Sadri, F., Mercaşand, R. Scott, G., A generalization of Thue freeness for partial words. Theoret. Comput. Sci. 410 (2009) 793–800. Google Scholar
F. Blanchet-Sadri, K. Black and A. Zemke, Unary pattern avoidance in partial words dense with holes. In Proc. of Language and Automata Theory and Applications – 5th International Conference, LATA 2011, Tarragona, Spain, May 26-31, 2011. Edited by A.H. Dediu, S. Inenaga and C. Martín-Vide. Vol. 6638 of Lect. Notes Comput. Science. Springer (2011) 155–166.
Dekking, F.M., Strongly nonrepetitive sequences and progression-free sets. J. Combin. Theory Ser. A 27 (1979) 181–185. Google Scholar
Erdős, P., Some unsolved problems. Magyar Tudományos Akadémia Matematikai Kutató Intézete 6 (1961) 221–254. Google Scholar
Huova, M., Existence of an infinite ternary 64-abelian square-free word. RAIRO: ITA 48 (2014) 307–314. Google Scholar
Huova, M. and Karhumäki, J., On the unavoidability of k-abelian squares in pure morphic words. J. Integer Seq. 16 (2013). Google Scholar
Huova, M., Karhumäki, J. and Saarela, A., Problems in between words and abelian words: k-abelian avoidability. Theoret. Comput. Sci. 454 (2012) 172–177. Google Scholar
M. Huova, J. Karhumäki, A. Saarela and K. Saari, Local squares, periodicity and finite automata. In Rainbow of Computer Science, edited by C. Calude, G. Rozenberg and A. Salomaa. Springer (2011) 90–101.
Karhumäki, J., Puzynina, S. and Saarela, A., Fine and Wilf’s theorem for k-abelian periods. Internat. J. Found. Comput. Sci. 24 (2013) 1135–1152. Google Scholar
V. Keränen, Abelian squares are avoidable on 4 letters. In Proc. of the 19th International Colloquium on Automata, Languages and Programming (1992) 41–52.
Manea, F. and Mercaş, R., Freeness of partial words. Theoret. Comput. Sci. 389 (2007) 265–277. Google Scholar
R. Mercaşand A. Saarela, 3-abelian cubes are avoidable on binary alphabets. In Proc. of the 17th International Conference on Developments in Language Theory, edited by M.-P. Béal and O. Carton, vol. 7907 of Lect. Notes Comput. Science. Springer (2013) 374–383.
M. Rao, On some generalizations of abelian power avoidability. preprint (2013).
A. Thue, Über unendliche Zeichenreihen. Norske Vid. Selsk. Skr. I, Mat. Nat. Kl. Christiania 7 (1906) 1–22. (Reprinted in Selected Mathematical Papers of A. Thue, T. Nagell, editor, Universitetsforlaget, Oslo, Norway (1977) 139–158).
A. Thue, Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Norske Vid. Selsk. Skr. I, Mat. Nat. Kl. Christiania 1 (1912) 1–67. (Reprinted in Selected Mathematical Papers of Axel Thue, T. Nagell, editor, Universitetsforlaget, Oslo, Norway (1977) 413–478).