Periodicity and Unbordered Words: A Proof of the Extended Duval Conjecture

TL;DR

This paper proves a stronger version of Duval's conjecture relating word length, unbordered factors, and periods, providing tight bounds and improving previous results in combinatorics on words.

Contribution

It establishes that for words with an unbordered prefix of certain length, the maximum unbordered factor length equals the period, solving a long-standing conjecture with tight bounds.

Findings

Proves f(w) = p(w) under specific conditions

Provides tight bounds for the relationship between word length and unbordered factors

Improves bounds related to the Ehrenfeucht and Silberger question

Abstract

The relationship between the length of a word and the maximum length of its unbordered factors is investigated in this paper. Consider a finite word w of length n. We call a word bordered, if it has a proper prefix which is also a suffix of that word. Let f(w) denote the maximum length of all unbordered factors of w, and let p(w) denote the period of w. Clearly, f(w) < p(w)+1. We establish that f(w) = p(w), if w has an unbordered prefix of length f(w) and n > 2f(w)-2. This bound is tight and solves the stronger version of a 21 years old conjecture by Duval. It follows from this result that, in general, n > 3f(w)-3 implies f(w) = p(w) which gives an improved bound for the question asked by Ehrenfeucht and Silberger in 1979.