Abstract

We characterize the formulas that are avoided by every $\alpha$-free word for some $\alpha>1$. We show that the avoidable formulas whose fragments are of the form $XY$ or $XYX$ are $4$-avoidable. The largest avoidability index of an avoidable palindrome pattern is known to be at least $4$ and at most $16$. We make progress toward the conjecture that every avoidable palindrome pattern is $4$-avoidable.

Highlights

  • IntroductionThe 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

  • 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

Read more

Summary

Introduction

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. An occurrence of a formula f in a word w is a non-erasing morphism h : ∆∗ → Σ∗ such that the h-image of every fragment of f is a factor of w. As for patterns, the index λ(f ) of a formula f is the size of the smallest alphabet allowing the existence of an infinite word containing no occurrence of f. A formula f is nice if for every variable X of f , there exists a fragment of f that contains X at least twice.

Preliminaries
Nice formulas
The avoidability exponent
Avoiding a nice formula
The number of fragments of a minimally avoidable formula
Characterization of some famous morphic words
Palindrome patterns
Discussion
Full Text
Paper version not known

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.