Extremal Square-Free Words

Abstract

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]

Extremal Overlap-Free and Extremal $\beta$-Free Binary Words

“…By Theorem 4, we know that ERT(2) 8/3. From the work of Grytczuk et al [11], we know that ERT(3) 2. It may be the case that ERT(2) = 8/3 and ERT(3) = 2, but we have only weak computational evidence supporting this conjecture.…”

“…The concept of extremal square-free word was recently introduced by Grytczuk et al [11], who demonstrated that there are arbitrarily long extremal square-free words over a ternary alphabet. Two of the present authors [14] adapted their ideas to find all lengths admitting extremal square-free ternary words.…”

“…In this paper, we consider some variations of extremal square-free words, with a focus on the binary alphabet Σ 2 = {0, 1}. We begin by considering extremal overlap-free words, as suggested by Grytczuk et al [11]. For a word w over a fixed alphabet Σ, we say that w is extremal overlap-free if w is overlap-free, and every extension of w contains an overlap.…”

The lengths for which bicrucial square-free permutations exist

“…The permutations can be seen in Figure 4 in Appendix B. 34 : (8,2,0,21,30,20,3,4,7,5,1,31,32,22,6,9,23,12,10,25,27,24,13,14,18,15,11,28,29,26,17,19,33,16) 36 : (27, 33, 35, 22, 21, 26, 32, 31, 28, 30, 34, 19, 12, 20, 29, 25, 11, 18, 24, 9, 7, 10, 17, 16, 13, 15, 23, 4, 2, 5, 14, 8, 1, 3, 6, 0) 40 : (8,2,0,20,29,19,3,4,7,5,1,30,31,21,6,9,22,12,10,24,26,<...>…”

“…Recently, the notion of bicrucial square-free words has been extended to the notion of extremal words, words which are square-free but inserting any letter in any position introduces a square [4], and this naturally extends to extremal permutations. Since square-free permutations must follow the up-up-down-down pattern, any insertion which breaks this pattern must introduce a square and so any insertion except in positions 0, 1, n − 1 or n introduces a square.…”