On ternary square-free circular words (Q612919)

Summary: 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.