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On the Tree of Ternary Square-Free Words

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Combinatorics on Words (WORDS 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9304))

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  • International Conference on Combinatorics on Words

Abstract

We contribute to the study of the set of ternary square-free words. Under the prefix order, this set forms a tree, and we prove two results on its structure. First, we show that non-branching paths in this tree are short: such a path from a node of nth level has length \(O(\log n)\). Second, we prove that any infinite path in the tree has a lot of branching points: the lower density of the set of such points is at least 2/9.

E.A. Petrova and A.M. Shur—Supported by the grant 13-01-00852 of the Russian Foundation of Basic Research.

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Notes

  1. 1.

    For big n, the bound can be lowered to \(8.5\log n-28\) by some additional argument.

  2. 2.

    The original proof contained a flaw, later fixed by Currie (see [2, 3]).

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Acknowledgement

We thank the anonymous referee for useful comments.

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Authors and Affiliations

  1. Ural Federal University, Ekaterinburg, Russia

    Elena A. Petrova & Arseny M. Shur

Authors
  1. Elena A. Petrova
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  2. Arseny M. Shur
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Correspondence to Elena A. Petrova .

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Editors and Affiliations

  1. Universität Kiel, Kiel, Germany

    Florin Manea

  2. Universität Kiel, Kiel, Germany

    Dirk Nowotka

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Petrova, E.A., Shur, A.M. (2015). On the Tree of Ternary Square-Free Words. In: Manea, F., Nowotka, D. (eds) Combinatorics on Words. WORDS 2015. Lecture Notes in Computer Science(), vol 9304. Springer, Cham. https://doi.org/10.1007/978-3-319-23660-5_19

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  • DOI: https://doi.org/10.1007/978-3-319-23660-5_19

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-23659-9

  • Online ISBN: 978-3-319-23660-5

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