Cycles of relatively prime length and the road coloring problem

We give a partial answer to theroad coloring problem, a purely graphtheoretical question with applications in both symbolic dynamics and automata theory. The question is whether for any positive integerk and for any aperiodic and strongly connected graphG with all vertices of out-degreek, we can labelG with symbols in an alphabet ofk letters so that all the edges going out from a vertex take a different label and all paths inG presenting a wordW terminate at the same vertex, for someW. Such a labelling is calledsynchronizing coloring ofG. Anyaperiodic graphG contains a setS of cycles where the greatest common divisor of the lengths equals 1. We establish some geometrical conditions onS to ensure the existence of a synchronizing coloring.