A note on conjectures generalizing the road colouring theorem

TL;DR

This paper explores conjectures extending the road colouring theorem to graphs with non-constant out-degree, providing theoretical proofs for specific graph classes and supporting evidence through computer simulations.

Contribution

It introduces two conjectures generalizing the road colouring theorem and proves them for certain classes of graphs, advancing understanding in graph synchronization.

Findings

Proved both conjectures for one class of graphs.

Proved one conjecture for an additional class.

Provided empirical evidence via computer simulations.

Abstract

The road colouring theorem characterizes the class of strongly connected directed graphs with constant out-degree that admit a synchronizing road colouring. The subject of this paper is a pair of related conjectures that generalize the road colouring theorem to graphs with non-constant out-degree; we give reasons to believe that both of these conjectures are true. Our main results focus on two classes of graphs, proving both conjectures for one class of graphs and one of the conjectures for an additional class of graphs. We also present computer simulations that give some empirical evidence for the conjectures.