## On ternary square-free circular words

### BibTeX

@MISC{Shur_onternary,

author = {Arseny M. Shur},

title = {On ternary square-free circular words},

year = {}

}

### OpenURL

### Abstract

Circular words are cyclically ordered finite sequences of letters. We give a computer-free proof of the following result by Currie: square-free circular words over the ternary alphabet exist for all lengths l except for 5, 7, 9, 10, 14, and 17. Our proof reveals an interesting connection between ternary square-free circular words and closed walks in the K3,3 graph. In addition, our proof implies an exponential lower bound on the number of such circular words of length l and allows one to list all lengths l for which such a circular word is unique up to isomorphism.

### Citations

647 |
Algebraic combinatorics on words
- Lothaire
- 2002
(Show Context)
Citation Context ...es a square-free ternary circular word of length l. Finally we describe a way to construct such walks of any given weight l � 18. 1 Preliminaries We recall some notation and definitions on words, see =-=[5]-=- for more background. An alphabet Σ is a nonempty finite set, the elements of which are called letters. Words are finite sequences of letters. As usual, we write Σ ∗ [Σ + ] for the set of all words ov... |

41 |
Sur un théorème de Thue
- Dejean
- 1972
(Show Context)
Citation Context ... bc... ˆw = 1 0 1110 . . . This type of encoding for an arbitrary k-ary k−1-free word was suggested by Pansiot [6] k−2 as a tool to operate with famous Dejean’s conjecture on avoidable exponents (see =-=[3]-=-). Pansiot’s encoding can be easily extended to circular words: any ternary square-free circular word w can be encoded, up to isomorphism, by a binary circular codeword ˆw of the same length l � 3, as... |

23 |
Nonrepetitive colorings of graphs, Random Struct. Alg
- Alon, Grytczuk, et al.
(Show Context)
Citation Context ...r, Proposition 1 below binds minimal squares and square-free circular words. the electronic journal of combinatorics 17 (2010), #R140 1The problem we study here has a graph theory origin. Namely, in =-=[1]-=- the existence of non-repetitive walks in coloured graphs was studied. The authors conjectured 1 which cycles Cl can be 3-coloured such that the label of any path is square-free. The existence of such... |

17 |
A propos d’une conjecture de F. Dejean sur les répétitions dans les mots
- Pansiot
- 1984
(Show Context)
Citation Context ...le is as follows: { 0, if w(i+2) = w(i), ˆw(i) = 1, otherwise. For example, w = ab c b ac bc... ˆw = 1 0 1110 . . . This type of encoding for an arbitrary k-ary k−1-free word was suggested by Pansiot =-=[6]-=- k−2 as a tool to operate with famous Dejean’s conjecture on avoidable exponents (see [3]). Pansiot’s encoding can be easily extended to circular words: any ternary square-free circular word w can be ... |

16 | There are ternary circular square-free words of length n for n ≥ 18,” Electronic
- Currie
- 2002
(Show Context)
Citation Context ...h is square-free. The existence of such a colouring is obviously equivalent to the existence of a circular square-free word of length l over the ternary alphabet. This conjecture was proved by Currie =-=[2]-=-: Theorem 1. Square-free circular words over the ternary alphabet exist for all lengths l except for l = 5, 7, 9, 10, 14, 17. The proof given by Currie consists of a construction allowing one to obtai... |

2 |
Nonrepetitive colorings of graphs – a survey
- Grytczuk
(Show Context)
Citation Context ...p the ends of the word w. 1 Currie [2] mentioned that this conjecture is originally due to R. J. Simpson but provided no reference. More details on non-repetitive colourings of graphs can be found in =-=[4]-=-. the electronic journal of combinatorics 17 (2010), #R140 2There is an obvious one-to-one correspondence between circular words and conjugacy classes of ordinary words. Thus, the definition of β-fre... |

1 |
Two-sided bounds for the growth rates of power-free languages
- Shur
(Show Context)
Citation Context ...e circular words is approximately 1.3. This growth rate obviously cannot exceed the growth rate for the set of ternary squarefree ordinary words. For the latter one, a nearly exact value is now known =-=[7]-=-. It is 1.30176..., so it is quite intriguing whether the two considered growth rates coincide. Our proof can also be used to list the lengths for which the ternary square-free circular word is unique... |