On ternary square-free circular words

TL;DR

This paper provides a computer-free proof of the existence of ternary square-free circular words for most lengths, reveals a connection to graph theory, and establishes bounds on their quantity and uniqueness.

Contribution

It offers a novel, computer-free proof of a known result, linking ternary square-free circular words to closed walks in the $K_{3,3}$ graph, and derives bounds on their counts.

Findings

Existence of ternary square-free circular words for all lengths except 5, 7, 9, 10, 14, 17.

An exponential lower bound on the number of such words for each length.

Identification of lengths with unique circular words up to isomorphism.

Abstract

Circular words are cyclically ordered finite sequences of letters. We give a computer-free proof of the following result by Currie: square-free circular words over the ternary alphabet exist for all lengths $l$ except for 5, 7, 9, 10, 14, and 17. Our proof reveals an interesting connection between ternary square-free circular words and closed walks in the $K_{3{,}3}$ graph. In addition, our proof implies an exponential lower bound on the number of such circular words of length $l$ and allows one to list all lengths $l$ for which such a circular word is unique up to isomorphism.