A Connection Between Unbordered Partial Words and Sparse Rulers - Saarela, Aleksi; Vanhatalo, Aleksi (The Electronic Journal of Combinatorics, 2026)
Partial words are words that contain, in addition to letters, special symbols called holes. Two partial words a=a_0 … a_n and b=b_0 … b_n are compatible if, for all i, a_i = b_i or at least one of a_i, b_i is a hole. A partial word is unbordered if it does not have a proper nonempty prefix and a suffix that are compatible. Otherwise, the partial word is bordered. A set R {0, …, n} is called a complete sparse ruler of length n if, for all k {0, …, n}, there exists r, s R such that k = r - s. These are also known as restricted difference bases. From the definitions it follows that the more holes a partial word has, the more likely it is to be bordered. By introducing a connection between unbordered partial words and sparse rulers, we improve the bounds on the maximum number of holes an unbordered partial word can have over alphabets of sizes 4 or greater. We also provide a counterexample for a previously reported theorem, depending on the reported values of the minimal number of marks in a length-135 ruler. We have not verified this value. We then study a two-dimensional generalization of these results. We adapt methods from the one-dimensional case to solve the correct asymptotic for the number of holes that an unbordered two-dimensional binary partial word can have.