### Periodicity and unbordered words: A proof of the extended duval conjecturePeriodicity and unbordered words: A proof of the extended duval conjecture

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 Author Harju, Tero ♦ Nowotka, Dirk Source ACM Digital Library Content type Text Publisher Association for Computing Machinery (ACM) File Format PDF Copyright Year ©2007 Language English
 Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science Subject Keyword Combinatorics on words ♦ Duval's conjecture ♦ Periodicity ♦ Unbordered words Abstract The relationship between the length of a word and the maximum length of its unbordered factors is investigated in this article. Consider a finite word $\textit{w}$ of length $\textit{n}.$ We call a word $\textit{bordered}$ if it has a proper prefix, which is also a suffix of that word. Let $μ(\textit{w})$ denote the maximum length of all unbordered factors of $\textit{w},$ and let $∂(\textit{w})$ denote the period of $\textit{w}.$ Clearly, $μ(\textit{w})$ ≤ $∂(\textit{w}).$ We establish that $μ(\textit{w})$ = $∂(\textit{w}),$ if $\textit{w}$ has an unbordered prefix of length $μ(\textit{w})$ and $\textit{n}$ ≥ $2μ(\textit{w})$ ™ 1. This bound is tight and solves the stronger version of an old conjecture by Duval [1983]. It follows from this result that, in general, $\textit{n}$ ≥ $3μ(\textit{w})$ ™ 3 implies $μ(\textit{w})$ = $∂(\textit{w}),$ which gives an improved bound for the question raised by Ehrenfeucht and Silberger in 1979. ISSN 00045411 Age Range 18 to 22 years ♦ above 22 year Educational Use Research Education Level UG and PG Learning Resource Type Article Publisher Date 2007-07-01 Publisher Place New York e-ISSN 1557735X Journal Journal of the ACM (JACM) Volume Number 54 Issue Number 4

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Source: ACM Digital Library