Abstract
Finite words and their overlap properties are considered in this paper. Let w be a finite word of length n with period p and where the maximum length of its unbordered factors equals k. A word is called unbordered if it possesses no proper prefix that is also a suffix of that word. Suppose k < p in w. It is known that n ≤ 2k − 2, if w has an unbordered prefix u of length k. We show that, if n = 2k − 2 then u ends in ab i, with two different letters a and b and i ≥ 1, and b i occurs exactly once in w. This answers a conjecture by Harju and the second author of this paper about a structural property of maximum Duval extensions. Moreover, we show here that i < k/3, which in turn leads us to the solution of a special case of a problem raised by Ehrenfeucht and Silberger in 1979.
The work on this article has been supported by the research project MSM 0021620839.
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References
Assous, R., Pouzet, M.: Une caractérisation des mots périodiques. Discrete Math. 25(1), 1–5 (1979)
Crochemore, M., Perrin, D.: Two-way string-matching. J. ACM 38(3), 651–675 (1991)
Duval, J.-P.: Relationship between the period of a finite word and the length of its unbordered segments. Discrete Math. 40(1), 31–44 (1982)
Duval, J.-P., Harju, T., Nowotka, D.: Unbordered factors and Lyndon words. Discrete Math. 308(11), 2261–2264 (2008)
Ehrenfeucht, A., Silberger, D.M.: Periodicity and unbordered segments of words. Discrete Math. 26(2), 101–109 (1979)
Harju, T., Nowotka, D.: Duval’s conjecture and Lyndon words. TUCS Tech. Rep. 479, Turku Centre of Computer Science, Finland (2002)
Harju, T., Nowotka, D.: Minimal Duval extensions. Internat. J. Found. Comput. Sci. 15(2), 349–354 (2004)
Harju, T., Nowotka, D.: Periodicity and unbordered words. In: Diekert, V., Habib, M. (eds.) STACS 2004. LNCS, vol. 2996, pp. 294–304. Springer, Heidelberg (2004)
Harju, T., Nowotka, D.: Periodicity and unbordered words: A proof of the extended Duval conjecture. J. ACM 54(4) (2007)
Holub, Š.: A proof of the extended Duval’s conjecture. Theoret. Comput. Sci. 339(1), 61–67 (2005)
Mignosi, F., Zamboni, L.Q.: A note on a conjecture of Duval and Sturmian words. Theor. Inform. Appl. 36(1), 1–3 (2002)
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Holub, Š., Nowotka, D. (2008). On the Relation between Periodicity and Unbordered Factors of Finite Words. In: Ito, M., Toyama, M. (eds) Developments in Language Theory. DLT 2008. Lecture Notes in Computer Science, vol 5257. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85780-8_32
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DOI: https://doi.org/10.1007/978-3-540-85780-8_32
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