Freeness of partial words

The paper approaches the classical combinatorial problem of freeness of words, in the more general case of partial words. First, we propose an algorithm that tests efficiently whether a partial word is k-free or not, for a given k. Then, we show that there exist arbitrarily many k-free infinite partial words, over a binary alphabet, containing an infinite number of holes, for k≥3. Moreover, we present an efficient algorithm for the construction of a cube-free partial word with a given number of holes, over a binary alphabet. In the final section of the paper, we show that there exists an infinite word, over a four-symbol alphabet, in which we can substitute randomly one symbol with a hole, and still obtain a cube-free word; we show that such a word does not exist for alphabets with fewer symbols. Further, we prove that in this word we can replace arbitrarily many symbols with holes, such that each two consecutive holes are separated by at least two symbols, and obtain a cube-free partial word. This result seems interesting because any partial word containing two holes with less than two symbols between them is not cube-free. Finally, we modify the previously presented algorithm to construct, over a four-symbol alphabet, a cube-free partial word with exactly n holes, having minimal length, among all the possible cube-free partial words with at least n holes.