Abstract
A word is square-free if it does not contain nonempty factors of the form $XX$. In 1906 Thue proved that there exist arbitrarily long square-free words over a $3$-letter alphabet. We consider a new type of square-free words with additional property. A square-free word is called extremal if it cannot be extended to a new square-free word by inserting a single letter at any position. We prove that there exist infinitely many square-free extremal words over a $3$-letter alphabet. Some parts of our construction relies on computer verifications. It is not known if there exist any extremal square-free words over a $4$-letter alphabet.
Highlights
A square is a nonempty word of the form XX
To construct extremal words of length exceeding any given constant it is enough to take a sufficiently long square-free walk W in DN∗ starting at Q and ending in R
Our computer experiment failed in finding extremal words over four letters of length up to 100
Summary
A square is a nonempty word of the form XX. For instance, aa, abab, abcabc, abacabac are examples of squares. There exist ternary square-free words of any length, as proved by Thue in [9] (see [3]). This result is the starting point of Combinatorics on Words, a wide discipline with lots of exciting problems, deep results, and important applications (see [1, 2, 4, 6, 7, 8]). There exist infinitely many extremal square-free words over a 3-letter alphabet. The proof is by recursive construction whose validity is partially based on computer verifications
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