Avoidability of circular formulas

Abstract

Clark has defined the notion of n-avoidance basis which contains the avoidable formulas with at most n variables that are closest to be unavoidable in some sense. The family C i of circular formulas is such that C 1 = AA, C 2 = ABA.BAB, C 3 = ABCA.BCAB.CABC and so on. For every i n, the n-avoidance basis contains C i . Clark showed that the avoidability index of every circular formula and of every formula in the 3-avoidance basis (and thus of every avoidable formula containing at most 3 variables) is at most 4. We determine exactly the avoidability index of these formulas.

Avoidability of Palindrome Patterns

“…All the nice formulas considered so far in the literature are also 3-avoidable. This includes doubled patterns [12], circular formulas [9], the nice formulas in the 3-avoidance basis [9], and the minimally nice ternary formulas in Table 1 [15]. Theorem 7 ([9,15]).…”

Avoidance bases for formulas with reversal

“…This means that no classical avoidable formula on at most three variables has avoidability index greater than 4. Recently, the exact avoidability index of every 3-minimal formula has been determined [9].…”

“…The formula xyzyx ⋅ zyxyz was proven to be 2-avoidable in [9], where it was shown that it does not occur in the image of any 5 4 + -free word over 5 letters under a particular 15uniform morphism m 15 ; see [9] for details. Unfortunately, the other members of Φ 3 that flatten to xyzyx ⋅ zyxyz all occur in the m 15 -image of some 5 4 + -free word (some prefix of the infinite 5 4 + -free word described by Moulin-Ollagnier [11], in fact).…”

Avoiding conjugacy classes on the 5-letter alphabet

“…Together with the construction of a morphic binary word avoiding every conjugacy class of length at least 5 and a morphic ternary word avoiding every conjugacy class of length at least 3 [1], this settles the topic of the smallest alphabet needed to avoid every conjugacy class of length at least k.…”

“…Bell and Madill [2] obtained a pure morphic word with this property (and some additional properties) over the 12-letter alphabet. Gamard et al [1] independently obtained a morphic word over the 6-letter alphabet. They also conjectured that the alphabet size can be lowered to 5, which is best possible.…”