A word is square-free if it does not contain non-empty factors of the form XX.In 1906 Thue proved that there exist arbitrarily long square-free words over 3-letter alphabet. We consider a new type of square-free words. A square-free word is extremal if it cannot be extended to a new square-free word by inserting a single letter on arbitrary position. We prove that there exist infinitely many extremal words over 3-letter alphabet. Some parts of our construction relies on computer verifications. We also pose some related open problems. arXiv:1910.06226v1 [math.CO]
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.It was proved recently [8] that among these words there are infinitely many extremal ones, that is, having a square in every single-letter extension.We study diverse problems concerning extensions of words preserving the property of avoiding squares. Our main motivation is the conjecture stating that there are no extremal words over a 4-letter alphabet. We also investigate a natural recursive procedure of generating square-free words by a single-letter right-most extension. We present the results of computer experiments supporting a supposition that this procedure gives an infinite squarefree word over any alphabet of size at least three.
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