Disposability in Square-Free Words

TL;DR

This paper investigates the existence of irreducibly square-free words over a three-letter alphabet, showing that such words exist for all lengths except 4, 5, 7, and 12, highlighting their structural properties.

Contribution

It demonstrates the existence and non-existence of irreducibly square-free words of specific lengths over a ternary alphabet, advancing understanding of their combinatorial structure.

Findings

Irreducibly square-free words exist for all lengths except 4, 5, 7, and 12.

Removing any internal letter from these words introduces a square.

The study characterizes the lengths for which such words can or cannot exist.

Abstract

We consider words $w$ over the alphabet $\Sigma=\{0,1,2\}$. It is shown that there are irreducibly square-free words of all lengths $n$ except 4,5,7 and 12. Such a word is square-free (i.e., it has no repetitions $uu$ as factors), but by removing any one internal letter creates a square in the word.