Decision Version of the Road Coloring Problem Is NP-Complete

After Trahtman in his brilliant paper [10] solved the Road Coloring Problem, a couple of new problems have arisen in the field of synchronizing automata. Some of them naturally extends questions related to the 'classical' version of synchronization. Particulary, it is known that the problem of finding the synchronizing word of a given length for a given automaton is NP-complete. Volkov [11] asked, what is the complexity of the following problem: given a constant out-degree digraph (possibly with multiple edges) and a natural number m, does there exist a synchronizing word of length in for some synchronizing labeling of G. In this paper we show that this decision version of the Road Coloring Problem is NP-complete.