Greedy palindromic lengths

TL;DR

This paper investigates the relationship between bounded greedy palindromic lengths of words and their periodicity, proving the conjecture that such words are ultimately periodic in specific cases.

Contribution

It introduces variants of the palindromic length conjecture, including greedy palindromic lengths, and proves the conjecture for these cases.

Findings

Bounded greedy palindromic lengths imply ultimate periodicity.

Introduces left and right greedy palindromic lengths.

Proves the conjecture in particular cases.

Abstract

In [A. Frid, S. Puzynina, L.Q. Zamboni, \textit{On palindromic factorization of words}, Adv. in Appl. Math. 50 (2013), 737-748], it was conjectured that any infinite word whose palindromic lengths of factors are bounded is ultimately periodic. We introduce variants of this conjecture and prove this conjecture in particular cases. Especially we introduce left and right greedy palindromic lengths. These lengths are always greater than or equals to the initial palindromic length. When the greedy left (or right) palindromic lengths of prefixes of a word are bounded then this word is ultimately periodic.