On ternary square-free circular words

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 K3,3 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.

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Arseny M. Shur
• Keywords: Circular Word, Computer-free Proof, CORR 2010, Education, Square-free Circular Words