Lengths of Irreducible and Delicate Words

TL;DR

This paper classifies the lengths of special binary and ternary words that are on the edge of repetition, focusing on irreducible and delicate words that are just barely free of squares, overlaps, or cubes.

Contribution

It provides a comprehensive classification of the lengths of irreducible and delicate words in the context of repetition-avoiding words over small alphabets.

Findings

Classified lengths of irreducible squarefree, overlap-free, cubefree words.

Classified lengths of delicate squarefree, overlap-free, cubefree words.

Results over binary and ternary alphabets.

Abstract

We study words that barely avoid repetitions, for several senses of "barely". A squarefree (respectively, overlap-free, cubefree) word is irreducible if removing any one of its interior letters creates a square (respectively, overlap, cube). A squarefree (respectively, overlap-free, cubefree) word is delicate if changing any one of its letters creates a square (respectively, overlap, cube). We classify the lengths of irreducible and delicate squarefree, overlap-free, and cubefree words over binary and ternary alphabets.