Independent sets of words and the synchronization problem

TL;DR

This paper explores the synchronization problem in locally strongly transitive automata, extending it to stable sets and minimal rank words, with applications to coloring aperiodic graphs with Hamiltonian paths.

Contribution

It introduces new extensions of the synchronization problem related to stable sets and minimal rank words, and applies these concepts to graph coloring problems.

Findings

Extended the synchronization problem to stable sets and minimal rank words.

Provided insights into coloring aperiodic graphs with Hamiltonian paths.

Analyzed properties of locally strongly transitive automata.

Abstract

The synchronization problem is investigated for the class of locally strongly transitive automata introduced in a previous work of the authors. Some extensions of this problem related to the notions of stable set and word of minimal rank of an automaton are studied. An application to synchronizing colorings of aperiodic graphs with a Hamiltonian path is also considered.