Abstract
We study extendability of ternary square-free words. Namely, we are interested in the square-free words that cannot be infinitely extended preserving square-freeness. We prove that any positive integer is the length of the longest extension of some ternary square-free word and thus solve an open problem by Allouche and Shallit. We also resolve the two-sided version of this problem.
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References
Allouche, J.P., Shallit, J.: Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press (2003)
Arshon, S.E.: Proof of the existence of asymmetric infinite sequences. Mat. Sbornik 2, 769–779 (1937) (in Russian, with French abstract)
Bean, D.A., Ehrenfeucht, A., McNulty, G.: Avoidable patterns in strings of symbols. Pacific J. Math. 85, 261–294 (1979)
Currie, J.D.: On the structure and extendibility of k-power free words. European J. Combinatorics 16, 111–124 (1995)
Klepinin, A.V., Sukhanov, E.V.: On combinatorial properties of the Arshon sequence. Diskretn. Anal. Issled. Oper. 6(6), 23–40 (1999) (in Russian); English translation in Disc. Appl. Math. 114, 155–169 (2001)
Pansiot, J.J.: A propos d’une conjecture de F. Dejean sur les répétitions dans les mots. Discrete Appl. Math. 7, 297–311 (1984)
Petrova, E.A., Shur, A.M.: Constructing premaximal binary cube-free words of any level. In: Proc. 8th Internat. Conf. Words 2011 (WORDS 2011). EPTCS, vol. 63, pp. 168–178 (2011)
Shur, A.M.: On ternary square-free circular words. Electronic J. Combinatorics 17(# R140) (2010)
Shur, A.M.: Deciding context equivalence of binary overlap-free words in linear time. Semigroup Forum 84, 447–471 (2012)
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Petrova, E.A., Shur, A.M. (2012). Constructing Premaximal Ternary Square-Free Words of Any Level. In: Rovan, B., Sassone, V., Widmayer, P. (eds) Mathematical Foundations of Computer Science 2012. MFCS 2012. Lecture Notes in Computer Science, vol 7464. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32589-2_65
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DOI: https://doi.org/10.1007/978-3-642-32589-2_65
Publisher Name: Springer, Berlin, Heidelberg
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