Synchronizing Road Coloring

The synchronizing word of a deterministic automaton is a word in the alphabet of colors (considered as letters) of its edges that maps the automaton to a single state. A coloring of edges of a directed graph is synchronizing if the coloring turns the graph into a deterministic finite automaton possessing a synchronizing word. The road coloring problem is the problem of synchronizing coloring of a directed finite strongly connected graph with constant outdegree of all its vertices if the greatest common divisor of lengths of all its cycles is one. The problem was posed by Adler, Goodwyn and Weiss over 30 years ago and evoked noticeable interest among the specialists in the theory of graphs, finite automata, coding and symbolic dynamics. Many partial solutions of the problem have been found and different generalizations were considered. The positive solution of the road coloring problem is presented below. We reproduce from the literature also the statements used in our proof. The necessary and sufficient conditions of synchronizing road coloring of directed graph with constant outdegree of a vertex are presented.