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 $K_{3{,}3}$ 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.

💡 Research Summary

The paper studies circular words over the ternary alphabet Σ = {a, b, c} that avoid squares, i.e., no factor of the form xx where x is a non‑empty word. A circular word is a finite sequence whose ends are identified, so the index arithmetic is performed modulo the length ℓ. The main theorem, originally proved by Currie with computer assistance, states that square‑free circular words exist for every length ℓ except ℓ ∈ {5, 7, 9, 10, 14, 17}. The authors present a completely elementary, computer‑free proof and, in the process, uncover a striking connection between such words and closed walks in the complete bipartite graph K₃,₃.

The proof begins by encoding each letter of Σ as a vertex on one side of K₃,₃ and each successive letter as a vertex on the opposite side. Consequently, each adjacent pair (w_i, w_{i+1}) of a circular word corresponds to an edge of K₃,₃, and the whole word corresponds to a closed walk of length ℓ. The square‑free condition translates into the requirement that the walk never traverses the same edge twice in a row; in graph‑theoretic language the walk must be “edge‑alternating” (no immediate edge repetitions). Thus the existence problem for square‑free circular words becomes the existence problem for edge‑alternating closed walks of a given length in K₃,₃.

The authors identify three elementary cycles in K₃,₃ that respect the edge‑alternating rule: a 4‑cycle (a → b → c → a), a 5‑cycle (a → b → c → b → a), and a 6‑cycle (a → b → c → a → b → c). By concatenating copies of these cycles, possibly inserting short “bridge” edges that preserve the alternation property, they can construct edge‑alternating closed walks of any length ℓ ≥ 11 except the six exceptional values. For each forbidden length they perform a case‑by‑case analysis showing that no combination of the basic cycles can close the walk without violating the alternation rule, thereby reproducing Currie’s list of exceptions without any exhaustive computer search.

Beyond existence, the construction yields quantitative information. Because each segment of length 6 can be independently chosen as either a 4‑cycle, a 5‑cycle, or a 6‑cycle, the number of distinct edge‑alternating closed walks (and hence square‑free circular words) grows at least as 2^{⌊ℓ/6⌋}. This provides an exponential lower bound on the counting function for ternary square‑free circular words, a result that was not explicit in Currie’s original work.

The paper also investigates the uniqueness question: for which lengths ℓ is there, up to rotation and reversal (the natural symmetries of a circular word), exactly one square‑free circular word? By examining the limited ways the basic cycles can be arranged for small ℓ, the authors compile a list of lengths where the construction is forced to a single pattern. These lengths include 11, 13, 15, and several larger values; the full list is presented in a table.

In the concluding section the authors discuss the broader significance of the graph‑theoretic viewpoint. The method suggests a template for tackling square‑free problems over larger alphabets by moving to K_{n,n} or other regular bipartite graphs, and it hints at possible extensions to avoid more complex repetitions such as cubes or fractional powers. The paper thus not only supplies a clean, non‑computational proof of Currie’s theorem but also opens a new avenue for combinatorial word research through the lens of graph walks.