On the Relation between Periodicity and Unbordered Factors of Finite Words

Finite words and their overlap properties are considered in this paper. Let w be a finite word of length n with period p and where the maximum length of its unbordered factors equals k. A word is called unbordered if it possesses no proper prefix that is also a suffix of that word. Suppose k < p in w. It is known that n ≤ 2k − 2, if w has an unbordered prefix u of length k. We show that, if n = 2k − 2 then u ends in ab, with two different letters a and b and i ≥ 1, and b occurs exactly once in w. This answers a conjecture by Harju and the second author of this paper about a structural property of maximum Duval extensions. Moreover, we show here that i < k/3, which in turn leads us to the solution of a special case of a problem raised by Ehrenfeucht and Silberger in 1979.