# Non-Repetitive Tilings

@article{Currie2002NonRepetitiveT, title={Non-Repetitive Tilings}, author={James D. Currie and Jamie Simpson}, journal={Electron. J. Comb.}, year={2002}, volume={9} }

In 1906 Axel Thue showed how to construct an infinite non-repetitive (or square-free) word on an alphabet of size 3. Since then this result has been rediscovered many times and extended in many ways. We present a two-dimensional version of this result. We show how to construct a rectangular tiling of the plane using 5 symbols which has the property that lines of tiles which are horizontal, vertical or have slope +1 or $-1$ contain no repetitions. As part of the construction we introduce a new…

## 24 Citations

### Infinite Sequences and Pattern Avoidance

- Mathematics
- 2004

The study of combinatorics on words dates back at least to the beginning of the 20 century and the work of Axel Thue. Thue was the first to give an example of an infinite word over a three letter…

### A word on 7 letters which is non-repetitive up to mod 5

- MathematicsActa Informatica
- 2003

A perturbed 6-cycle is constructed which is non-repetitive up to mod 5 and answers a question of Currie and Simpson about bisequences where certain subsequences are non- repetitive.

### On a conjecture on k-Thue sequences

- Mathematics
- 2016

A repetition in a sequence S is a subsequence ξ1 . . . ξtξt+1 . . . ξ2t of consecutive terms of S such that ξi = ξt+i for every i = 1, . . . , t. A sequence is called nonrepetitive or Thue if it does…

### Nonrepetitive colorings of graphs

- MathematicsRandom Struct. Algorithms
- 2002

This paper considers a natural generalization of Thue's sequences for colorings of graphs and shows that the Thue number stays bounded for graphs with bounded maximum degree, in particular, π (G) ≤ cΔ(G)2 for some absolute constant c.

### On a Generalization of Thue Sequences

- MathematicsElectron. J. Comb.
- 2015

A construction of k-Thue sequences on 2k symbols is presented, which improves the previous bound of $2k + 10\sqrt{k}$ and obtain tight bounds for total Thue colorings of cycles.

### Nonrepetitive Sequences on Arithmetic Progressions

- MathematicsElectron. J. Comb.
- 2011

It is proved that for every k and every c, there exist arbitrarily long sequences over at most $(1+\frac{1}{c})k+18k^{c/c+1}$ symbols whose subsequences indexed by arithmetic progressions with common differences from the set $\{1,2,...,k\}$ are nonrepetitive.

## References

SHOWING 1-10 OF 10 REFERENCES

### Avoidable patterns in strings of symbols

- Mathematics
- 1979

A word is just a finite string of letters. The word Wavoids the word U provided no substitution instance of Uis a subword of W. W is avoidable if on some finite alpha-bet there is an infinite…

### INFINITE PERIODIC GROUPS. I

- Mathematics
- 1968

In this paper we construct an example of an infinite periodic group with a finite number of generators, in which the orders of all the elements are bounded by a specified number. This is a solution…

### Über unendliche Zeichenreihen, Norske Vid

- Selsk. Skr. I. Mat. Nat. Kl. Christiana
- 1906

### Über unendliche Zeichenreihen, Norske Vid

- Selsk. Skr. I. Mat. Nat. Kl. Christiana
- 2002

### Hedlund , Symbolic dynamics I , II

- Amer . J . Math .

### Symbolic dynamics I, II, Amer

- J. Math
- 1938

### Selsk. Skr. I. Mat. Nat. Kl. Christiana

- Selsk. Skr. I. Mat. Nat. Kl. Christiana
- 1906

### the electronic journal of combinatorics

- the electronic journal of combinatorics
- 2002