Hostname: page-component-74656f9884-rdgjk Total loading time: 0 Render date: 2026-11-09T04:19:50.762Z 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. rgm@informatik.uni-kiel.de
Aleksi Saarela
Affiliation:
Department of Mathematics and Statistics, University of Turku, 20014 Turku, Finland; amsaar@utu.fi

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

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

Keywords

Combinatorics on wordsk-abelianequivalencerepetition-freenesscube-freeness

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).