Periodicity and Unbordered Words: A Proof of Duval?s Conjecture

The relationship between the length of a word and the maximum length of its unbordered factors is investigated in this paper. A word is bordered, if it has a proper prefix that is also a suffix of that word. Consider a finite word w of length n. Let μ(w) denote the maximum length of its unbordered factors, and let ∂(w) denote the period of w. Clearly, μ(w) ≤ ∂(w). We establish that μ(w) = ∂(w), if w has an unbordered prefix of length μ(w) and n ≥ 2μ(w)− 1. This bound is tight and solves a 21 year old conjecture by Duval. It follows from this result that, in general, n ≥ 3μ(w) implies μ(w) = ∂(w) which gives an improved bound for the question asked by Ehrenfeucht and Silberger in 1979.