There Are Ternary Circular Square-Free Words of Length $n$ for $n\ge 18$

Currie, James D.

doi:10.37236/1671

Cited by 63 publications

(51 citation statements)

References 7 publications

(4 reference statements)

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“…This graph has a facial nonrepetitive 6-coloring as follows. Use a nonrepetitive 3-coloring on the outer face with colors 1, 2, 3 [5]. After that use the colors 4, 5, 6 and 1, 2, 3 alternately to color the outer layer of the remaining graph as in the proof of Theorem 8.…”

Section: Factmentioning

confidence: 99%

Facial Nonrepetitive Vertex Coloring of Plane Graphs

Barát

Czap

2012

J. Graph Theory

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A sequence s 1 , s 2 , . . . , s k , s 1 , s 2 , . . . , s k is a repetition. A sequence S is nonrepetitive, if no subsequence of consecutive terms of S is a repetition. Let G be a plane graph. That is, a planar graph with a fixed embedding in the plane. A facial path consists of consecutive vertices on the boundary of a face. A facial nonrepetitive vertex coloring of a plane graph G is a vertex coloring such that the colors assigned to the vertices of any facial path form a nonrepetitive sequence. Let π f (G) denote the minimum

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Section: Factmentioning

confidence: 99%

Facial Nonrepetitive Vertex Coloring of Plane Graphs

Barát

Czap

2012

J. Graph Theory

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“…Several graph classes are known to have bounded nonrepetitive chromatic number. In particular, cycles are nonrepetitively 3-colourable (except for a finite number of exceptions) [12], trees are nonrepetitively 4-colourable [9,36], outerplanar graphs are nonrepetitively 12-colourable [5,36], and more generally, every graph with treewidth k is nonrepetitively 4 k -colourable [36]. Graphs with maximum degree ∆ are nonrepetitively O(∆ 2 )-colourable [3,17,27,32], and graphs excluding a fixed immersion have bounded nonrepetitive chromatic number [53].…”

Section: Introductionmentioning

confidence: 99%

Planar graphs have bounded nonrepetitive chromatic number

Dujmović

Esperet

Joret

et al. 2020

AiC

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A colouring of a graph is nonrepetitive if for every path of even order, the sequence of colours on the first half of the path is different from the sequence of colours on the second half. We show that planar graphs have nonrepetitive colourings with a bounded number of colours, thus proving a conjecture of Alon, Grytczuk, Hałuszczak and Riordan (2002). We also generalise this result for graphs of bounded Euler genus, graphs excluding a fixed minor, and graphs excluding a fixed topological minor.

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“…Almost all values of the strong circular repetition threshold CRT S (k) are known. Aberkane and Currie [1] demonstrated that CRT S (2) = 5/2, while the fact that CRT S (3) = 2 follows from the work of Currie [5] along with a finite search, or alternatively from the work of Shur [18]. Gorbunova [10] demonstrated that CRT S (k) = ⌈k/2⌉+1 ⌈k/2⌉ for all k ≥ 6, and conjectured that this formula holds for k = 4 and k = 5 as well.…”

Section: Introductionmentioning

confidence: 99%

Circular Repetition Thresholds on Some Small Alphabets: Last Cases of Gorbunova's Conjecture

Currie

Mol

Rampersad

2019

Electron. J. Combin.

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A word is called β-free if it has no factors of exponent greater than or equal to β. The repetition threshold RT(k) is the infimum of the set of all β such that there are arbitrarily long k-ary β-free words (or equivalently, there are k-ary β-free words of every sufficiently large length, or even every length). These three equivalent definitions of the repetition threshold give rise to three natural definitions of a repetition threshold for circular words. The infimum of the set of all β such that (a) there are arbitrarily long k-ary β-free circular words is called the weak circular repetition threshold, denoted CRTW(k);(b) there are k-ary β-free circular words of every sufficiently large length is called the intermediate circular repetition threshold, denoted CRTI(k);(c) there are k-ary β-free circular words of every length is called the strong circular repetition threshold, denoted CRTS(k).We prove that CRTS(4) = 3 2 and CRTS(5) = 4 3 , confirming a conjecture of Gorbunova and providing the last unknown values of the strong circular repetition threshold. We also prove that CRTI(3) = CRTW(3) = RT(3) = 7 4 . MSC 2010: 68R15

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There Are Ternary Circular Square-Free Words of Length $n$ for $n\ge 18$

Abstract: There are circular square-free words of length $n$ on three symbols for $n\ge 18$. This proves a conjecture of R. J. Simpson.

Cited by 63 publications

References 7 publications

Facial Nonrepetitive Vertex Coloring of Plane Graphs

Facial Nonrepetitive Vertex Coloring of Plane Graphs

Planar graphs have bounded nonrepetitive chromatic number

Circular Repetition Thresholds on Some Small Alphabets: Last Cases of Gorbunova's Conjecture

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