Non-repetitive Strings over Alphabet Lists

Mhaskar, Neerja; Soltys, Michael

doi:10.1007/978-3-319-15612-5_24

Cited by 3 publications

(3 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Is every path nonrepetitively 3-choosable? [43,69,108]? Note that a simple adaptation to the proof of Theorem 3.6 shows that every path is list 3-colourable such that every subpath with at least four vertices is nonrepetitively coloured; that is, the only repetitively coloured subpaths have two vertices.…”

Section: Theorem 36 ([131]mentioning

confidence: 99%

Nonrepetitive Graph Colouring

Wood

2021

Electron. J. Combin.

View full text Add to dashboard Cite

A vertex colouring of a graph $G$ is nonrepetitive if $G$ contains no path for which the first half of the path is assigned the same sequence of colours as the second half. Thue's famous theorem says that every path is nonrepetitively 3-colourable. This paper surveys results about nonrepetitive colourings of graphs. The goal is to give a unified and comprehensive presentation of the major results and proof methods, as well as to highlight numerous open problems.

show abstract

Section: Theorem 36 ([131]mentioning

confidence: 99%

Nonrepetitive Graph Colouring

Wood

2021

Electron. J. Combin.

View full text Add to dashboard Cite

show abstract

“…Is every path nonrepetitively list 3-colourable? [42,70,107]? Note that a simple adaptation to the proof of Theorem 3.6 shows that every path is list 3colourable such that every subpath with at least four vertices is nonrepetitively coloured; that is, the only repetitively coloured subpaths have two vertices.…”

Section: Theorem 36 ([130]mentioning

confidence: 99%

Nonrepetitive graph colouring

Wood

2020

Preprint

View full text Add to dashboard Cite

A vertex colouring of a graph G is nonrepetitive if G contains no path for which the first half of the path is assigned the same sequence of colours as the second half. Thue's famous theorem says that every path is nonrepetitively 3-colourable. This paper surveys results about nonrepetitive colourings of graphs. The goal is to give a unified and comprehensive presentation of the major results and proof methods, as well as to highlight numerous open problems.

show abstract

“…It was first conjectured by Grytczuk [6] that the nonrepetitive list chromatic number of any path is in fact at most 3. This conjecture has been mentioned many times, but not much progress has been made in the direction of proving or disproving it (see for instance [5,6,7,9,13,14,16,20,21] for some of the mentions of this problem).…”

Section: Introductionmentioning

confidence: 99%

Avoiding squares over words with lists of size three amongst four symbols

Rosenfeld¹

2021

Preprint

View full text Add to dashboard Cite

In 2007, Grytczuk conjecture that for any sequence (ℓ i ) i≥1 of alphabets of size 3 there exists a square-free infinite word w such that for all i, the i-th letter of w belongs to ℓ i . The result of Thue of 1906 implies that there is an infinite square-free word if all the ℓ i are identical. On the other, hand Grytczuk, Przyby lo and Zhu showed in 2011 that it also holds if the ℓ i are of size 4 instead of 3.In this article, we first show that if the lists are of size 4, the number of square-free words is at least 2.45 n (the previous similar bound was 2 n ). We then show our main result: we can construct such a square-free word if the lists are subsets of size 3 of the same alphabet of size 4. Our proof also implies that there are at least 1.25 n square-free words of length n for any such list assignment. This proof relies on the existence of a set of coefficients verified with a computer. We suspect that the full conjecture could be resolved by this method with a much more powerful computer (but we might need to wait a few decades for such a computer to be available).

show abstract

Non-repetitive Strings over Alphabet Lists

Cited by 3 publications

References 8 publications

Nonrepetitive Graph Colouring

Nonrepetitive Graph Colouring

Nonrepetitive graph colouring

Avoiding squares over words with lists of size three amongst four symbols

Contact Info

Product

Resources

About