[go: up one dir, main page]
More Web Proxy on the site http://driver.im/
Skip to main content
Springer Nature Link
Log in

On the Tree of Ternary Square-Free Words

  • Conference paper
  • First Online:
Combinatorics on Words (WORDS 2015)

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

Included in the following conference series:

  • 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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
GBP 19.95
Price includes VAT (United Kingdom)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
GBP 35.99
Price includes VAT (United Kingdom)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
GBP 44.99
Price includes VAT (United Kingdom)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

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]).

References

  1. Bean, D.A., Ehrenfeucht, A., McNulty, G.: Avoidable patterns in strings of symbols. Pac. J. Math. 85, 261–294 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  2. Currie, J.D.: On the structure and extendibility of \(k\)-power free words. Eur. J. Comb. 16, 111–124 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  3. Currie, J.D., Pierce, C.W.: The fixing block method in combinatorics on words. Combinatorica 23, 571–584 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Currie, J.D., Shelton, R.O.: The set of \(k\)-power free words over \(\sigma \) is empty or perfect. Eur. J. Comb. 24, 573–580 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Pansiot, J.J.: A propos d’une conjecture de F. Dejean sur les répétitions dans les mots. Discrete Appl. Math. 7, 297–311 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  6. Petrova, E.A., Shur, A.M.: Constructing premaximal binary cube-free words of any level. Int. J. Found. Comp. Sci. 23(8), 1595–1609 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Petrova, E.A., Shur, A.M.: Constructing premaximal ternary square-free words of any level. In: Rovan, B., Sassone, V., Widmayer, P. (eds.) MFCS 2012. LNCS, vol. 7464, pp. 752–763. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  8. Restivo, A., Salemi, S.: Some decision results on non-repetitive words. In: Apostolico, A., Galil, Z. (eds.) Combinatorial Algorithms on Words. NATO ASI Series, vol. 12, pp. 289–295. Springer, Heidelberg (1985)

    Chapter  Google Scholar 

  9. Shelton, R.: Aperiodic words on three symbols. II. J. Reine Angew. Math. 327, 1–11 (1981)

    MathSciNet  MATH  Google Scholar 

  10. Shelton, R.O., Soni, R.P.: Aperiodic words on three symbols. III. J. Reine Angew. Math. 330, 44–52 (1982)

    MathSciNet  MATH  Google Scholar 

  11. Shur, A.M.: On ternary square-free circular words. Electron. J. Comb. 17, R140 (2010)

    MathSciNet  Google Scholar 

  12. Shur, A.M.: Deciding context equivalence of binary overlap-free words in linear time. Semigroup Forum 84, 447–471 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Shur, A.M.: Growth properties of power-free languages. Comput. Sci. Rev. 6, 187–208 (2012)

    Article  MATH  Google Scholar 

  14. Thue, A.: Über unendliche Zeichenreihen. Norske vid. Selsk. Skr. Mat. Nat. Kl. 7, 1–22 (1906)

    Google Scholar 

Download references

Acknowledgement

We thank the anonymous referee for useful comments.

Author information

Authors and Affiliations

  1. Ural Federal University, Ekaterinburg, Russia

    Elena A. Petrova & Arseny M. Shur

Authors
  1. Elena A. Petrova
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Arseny M. Shur
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Elena A. Petrova .

Editor information

Editors and Affiliations

  1. Universität Kiel, Kiel, Germany

    Florin Manea

  2. Universität Kiel, Kiel, Germany

    Dirk Nowotka

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

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

Download citation

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

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-23659-9

  • Online ISBN: 978-3-319-23660-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics