Abstract
We construct an infinite word w over the 5-letter alphabet such that for every factor f of w of length at least two, there exists a cyclic permutation of f that is not a factor of w. In other words, w does not contain a non-trivial conjugacy class. This proves the conjecture in Gamard et al. [Theoret. Comput. Sci. 726 (2018) 1–4].
Highlights
A pattern p is a non-empty finite word over an alphabet ∆ = {A, B, C, . . .} of capital letters called variables
The avoidability index λ(p) of a pattern p is the size of the smallest alphabet Σ such that there exists an infinite word over Σ containing no occurrence of p
The infinite word F ω(0) contains only the conjugacy classes listed in C = F (2), F 2(2), F d(4), f d(0), for all d 1
Summary
The avoidability index λ(p) of a pattern p is the size of the smallest alphabet Σ such that there exists an infinite word over Σ containing no occurrence of p. Bell and Madill [3] obtained a pure morphic word over the 12-letter alphabet that satisfies P2 and some other properties. We prove this conjecture using a morphic word This settles the topic of the smallest alphabet needed to satisfy Pi
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