A unique extension of rich words

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∈R, let K(w) denote the set of all words u such that wu∈R and wu can be extended in at least two ways. Let ω(w)=min⁡{|u||u∈K(w)} and let ϕ(n)=max⁡{ω(w)|w∈R and |w|=n}, where n>0. Vesti (2014) showed that ϕ(n)≤2n. In other words, it says that for each w∈R there is a word u with |u|≤2|w| such that wu∈R and wu can be extended in at least two ways.We prove that ϕ(n)≤n and that limsupn→∞ϕ(n)n≥29. The results hold for each finite alphabet having at least two letters.