  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.
