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arXiv · 1009.5759

On ternary square-free circular words

Arseny M. Shur
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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.

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Arseny M. Shur. 2010-09-29. On ternary square-free circular words. https://arxiv.org/abs/1009.5759

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