Generalized de Bruijn words for Primitive words and Powers

TL;DR

This paper introduces a method to construct words with circular factors corresponding to primitive words, explores connections between de Bruijn and Lyndon graphs, and analyzes the minimal length of words containing all p-powers, providing algorithms for these constructions.

Contribution

It presents a new construction of words related to primitive words and powers, and establishes bounds and algorithms for generating such words over finite alphabets.

Findings

Existence of words with circular factors corresponding to primitive words for all n

Connections established between de Bruijn graphs of primitive words and Lyndon graphs

Bounds and algorithms for shortest words containing all p-powers of a given length

Abstract

We show that for every $n \geq 1$ and over any finite alphabet, there is a word whose circular factors of length $n$ have a one-to-one correspondence with the set of primitive words. In particular, we prove that such a word can be obtained by a greedy algorithm, or by concatenating all Lyndon words of length $n$ in increasing lexicographic order. We also look into connections between de Bruijn graphs of primitive words and Lyndon graphs. Finally, we also show that the shortest word that contains every $p$-power of length $pn$ over a $k$-letter alphabet has length between $pk^n$ and roughly $(p+ \frac{1}{k}) k^n$, for all integers $p \geq 1$. An algorithm that generates a word which achieves the upper bound is provided.