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On Some Interesting Ternary Formulas

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Combinatorics on Words (WORDS 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10432))

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Abstract

We show that, up to renaming of the letters, the only infinite ternary words avoiding the formula ABCAB.ABCBA.ACB.BAC (resp. ABCA.BCAB.BCB.CBA) have the same set of recurrent factors as the fixed point of \(\texttt {0}\mapsto \texttt {012}\), \(\texttt {1}\mapsto \texttt {02}\), \(\texttt {2}\mapsto \texttt {1}\).

Also, we show that the formula ABAC.BACA.ABCA is 2-avoidable. Finally, we show that the pattern ABACADABCA is unavoidable for the class of \(C_4\)-minor-free graphs with maximum degree 3. This disproves a conjecture of Grytczuk.

This work was partially supported by the ANR project CoCoGro (ANR-16-CE40-0005).

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References

  1. Berstel, J.: Axel Thue’s papers on repetitions in words: a translation, vol. 20. Publications du LACIM. Université du Québec à Montréal (1994)

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  3. Grytczuk, J.: Pattern avoidance on graphs. Discrete Math. 307(11–12), 1341–1346 (2007)

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  5. Thue, A.: Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Norske Vid. Selsk. Skr. I. Mat. Nat. Kl. Christiania 10, 1–67 (1912)

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Author information

Authors and Affiliations

  1. LIRMM, CNRS, Université de Montpellier, Montpellier, France

    Pascal Ochem

  2. LIP, ENS de Lyon, CNRS, UCBL, Université de Lyon, Lyon, France

    Matthieu Rosenfeld

Authors
  1. Pascal Ochem
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  2. Matthieu Rosenfeld
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Corresponding author

Correspondence to Pascal Ochem .

Editor information

Editors and Affiliations

  1. Université du Québec à Montréal, Montreal, Québec, Canada

    Srečko Brlek

  2. Université du Québec à Montréal, Montreal, Québec, Canada

    Francesco Dolce

  3. Université du Québec à Montréal, Montreal, Québec, Canada

    Christophe Reutenauer

  4. Université du Québec à Montréal, Montreal, Québec, Canada

    Élise Vandomme

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Ochem, P., Rosenfeld, M. (2017). On Some Interesting Ternary Formulas. In: Brlek, S., Dolce, F., Reutenauer, C., Vandomme, É. (eds) Combinatorics on Words. WORDS 2017. Lecture Notes in Computer Science(), vol 10432. Springer, Cham. https://doi.org/10.1007/978-3-319-66396-8_4

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  • DOI: https://doi.org/10.1007/978-3-319-66396-8_4

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-66395-1

  • Online ISBN: 978-3-319-66396-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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