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 theorem 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.