Extensions and reductions of square-free words

TL;DR

This paper explores special classes of square-free words, such as steady and bifurcate words, establishing their existence, properties, and bounds over various alphabet sizes, with implications for combinatorics on words.

Contribution

It introduces the concepts of steady and bifurcate square-free words, proves their existence over certain alphabet sizes, and conjectures bounds for minimal alphabet sizes needed.

Findings

Existence of infinitely many steady words over a 4-letter alphabet.

Construction of steady words of any length from 7-letter alphabets.

Every steady word is bifurcate over alphabets with at least three letters.

Abstract

A word is square-free if it does not contain a nonempty word of the form $XX$ as a factor. A famous 1906 result of Thue asserts that there exist arbitrarily long square-free words over a $3$-letter alphabet. We study square-free words with additional properties involving single-letter deletions and extensions of words. A square-free word is steady if it remains square-free after deletion of any single letter. We prove that there exist infinitely many steady words over a $4$-letter alphabet. We also demonstrate that one may construct steady words of any length by picking letters from arbitrary alphabets of size $7$ assigned to the positions of the constructed word. We conjecture that both bounds can be lowered to $4$, which is best possible. In the opposite direction, we consider square-free words that remain square-free after insertion of a single (suitably chosen) letter at every possible position in the word. We call them bifurcate. We prove a somewhat surprising fact, that over a fixed alphabet with at least three letters, every steady word is bifurcate. We also consider families of bifurcate words possessing a natural tree structure. In particular, we prove that there exists an infinite tree of doubly infinite bifurcate words over alphabet of size $12$.