Abstract
An overlap-free (or $\beta$-free) word $w$ over a fixed alphabet $\Sigma$ is extremal if every word obtained from $w$ by inserting a single letter from $\Sigma$ at any position contains an overlap (or a factor of exponent at least $\beta$, respectively). We find all lengths which admit an extremal overlap-free binary word. For every "extended" real number $\beta$ such that $2^+\leqslant\beta\leqslant 8/3$, we show that there are arbitrarily long extremal $\beta$-free binary words.
Highlights
Throughout, we use standard definitions and notation from combinatorics on words
A word is square-free if it contains no square as a factor, and overlap-free if it contains no overlap as a factor
We show that w is overlap-free
Summary
Throughout, we use standard definitions and notation from combinatorics on words (see [12]). The word abcabacbcabcbabcabacbcabc of length 25 is an extremal square-free word of minimum length over the alphabet {a, b, c}. We consider some variations of extremal square-free words, with a focus on the binary alphabet Σ2 = {0, 1}. Our first main result is the following characterization of the lengths of extremal overlap-free binary words. There is an extremal overlap-free word of length n over the alphabet Σ2 if and only if n is in the set. By Theorem 1, we know that there are arbitrarily long extremal 2+-free words over Σ2. Every binary word of length at least 4 contains a square, so it follows that for all β 2, there do not exist arbitrarily long extremal β-free words over Σ2. We conclude with a discussion of some open problems and conjectures over larger alphabets
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
