A quadratic algorithm for road coloring

The Road Coloring Theorem states that every aperiodic directed graph with constant out-degree has a synchronized coloring. This the-orem had been conjectured during many years as the Road Coloring Problem before being settled by A. Trahtman. Trahtman’s proof leads to an algorithm that finds a synchronized labeling with a cubic worst-case time complexity. We show a variant of his construction with a worst-case complexity which is quadratic in time and linear in space. We also extend the Road Coloring Theorem to the periodic case.