A Unique Extension of Rich Words

TL;DR

This paper studies the structure of rich words, showing that they can be extended in multiple ways within a linear bound, and establishes bounds on the minimal extension length needed for such properties.

Contribution

It improves the known upper bound on the extension length for rich words from 2n to n, and provides a lower bound proportional to n, for alphabets with at least two letters.

Findings

The upper bound on extension length is improved to n.

There exists a proportional lower bound for extension length, at least (2/9 - c)n.

Results apply to alphabets with at least two letters.

Abstract

A word $w$ is called rich if it contains $| w|+1$ palindromic factors, including the empty word. We say that a rich word $w$ can be extended in at least two ways if there are two distinct letters $x,y$ such that $wx,wy$ are rich. Let $R$ denote the set of all rich words. Given $w\in R$, let $K(w)$ denote the set of all words such that if $u\in K(w)$ then $wu\in R$ and $wu$ can be extended in at least two ways. Let $\omega(w)=\min\{| u| \mid u\in K(w)\}$ and let $\phi(n)=\max\{\omega(w)\mid w\in R\mbox{ and }| w|=n\}$, where $n>0$. Vesti (2014) showed that $\phi(n)\leq 2n$. In other words, it says that for each $w\in R$ there is a word $u$ with $| u|\leq 2| w|$ such that $wu\in R$ and $wu$ can be extended in at least two ways. We prove that $\phi(n)\leq n$. In addition we prove that for each real constant $c>0$ and each integer $m>0$ there is $n>m$ such that $\phi(n)\geq (\frac{2}{9}-c)n$. The results hold for each finite alphabet having at least two letters.