Forced Repetitions over Alphabet Lists
نویسندگان
چکیده
Thue [14] showed that there exist arbitrarily long square-free strings over an alphabet of three symbols (not true for two symbols). An open problem was posed in [7], which is a generalization of Thue’s original result: given an alphabet list L = L1, . . . , Ln, where |Li| = 3, is it always possible to find a square-free string, w = w1w2 . . .wn, where wi ∈ Li? In this paper we show that squares can be forced on square-free strings over alphabet lists iff a suffix of the square-free string conforms to a pattern which we term as an offending suffix. We also prove properties of offending suffixes. However, the problem remains tantalizingly open.
منابع مشابه
Square-free Strings Over Alphabet Lists
We solve an open problem that was posed in [GKM10]: Given an alphabet list L = L1, . . . , Ln, where |Li| = 3 and 0 ≤ i ≤ n, can we always find a square-free string, W = W1W2 . . .Wn, where Wi ∈ Li? We show that this is indeed the case. We do so using an “offending suffix” characterization of forced repetitions, and a counting, non-constructive, technique. We discuss future directions related t...
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