In the first paper, we investigate squares in partial words with one hole. A square in a partial word over a given alphabet has the form uv where u is compatible with v, and consequently, such square is compatible with a number of words over the alphabet that are squares. A well known result of Fraenkel and Simpson states that the number of distinct squares in a word of length n is bounded by 2n since at each position there are at most two distinct squares whose last occurrence starts. Recently, it was shown that for partial words with one hole, there may be more than two squares that have their last occurrence starting at the same position. Here, we prove that if such is the case, then the length of the shortest square is at most half the length of the third shortest square. As a result, we show that the number of distinct squares compatible with factors of a partial word with one hole of length n is bounded by 7n/2.
In the second paper, we extend the three-squares lemma on words to partial words with one hole. This result provides special information about the squares in a partial word with at most one hole, and puts restrictions on the positions at which periodic factors may occur, which is in contrast with the well known periodicity lemma of Fine and Wilf.
Keywords: Combinatorics on words; Partial words; Squares; Fine and Wilf's periodicity lemma; Three-squares lemma.