Road Coloring Problem

In the directed graph above, pick any vertex and follow the arrows in sequence blue-red-red three times. You will finish at the green vertex. Similarly, follow the sequence blue-blue-red three times and you will always end on the yellow vertex, no matter where you started. This is called a synchronized coloring.

The road coloring problem is the problem of synchronizing coloring of a directed finite strongly connected graph with the same outdegree and where the greatest common divisor of all cycles lengths is 1. Trahtman (2007) provided a positive solution to this problem.

REFERENCES

Adler, R. L.; Goodwyn, L. W.; Weiss, B. "Equivalence of Topological Markov Shifts." Israel J. Math. 27, 49-63, 1977.

Adler, R. L. and Weiss, B. Similarity of Automorphisms of the Torus. Providence, RI: Amer. Math. Soc., 1970.

Trahtman, A. N. "The Road Coloring Problem." 21 Dec 2007. http://arxiv.org/abs/0709.0099.