The Road Coloring for Mapping on k States

Let Γ be directed strongly connected finite graph of uniform outdegree (constant outdegree of any vertex) and let some coloring of edges of Γ turn the graph into deterministic complete automaton. Let the word s be a word in the alphabet of colors (considered also as letters) on the edges of Γ and let Γs be a mapping of vertices Γ. A coloring is called k-synchronizing if for any word t |Γt| ≥ k and for some word s (called k-synchronizing word) |Γs| = k. The k-synchronizing coloring turns the graph into a deterministic finite automaton possessing a k-synchronizing word. The road coloring problem is a problem of k-synchronizing coloring of Γ for k = 1. The problem was posed by Adler, Goodwyn andWeiss over 30 years ago and evoked noticeable interest. The recent positive solution of the road coloring problem instantly posed the problem of k-synchronizing coloring for arbitrary integer k. The necessary and sufficient conditions of k-synchronizing coloring are presented. They have the same simple form as in the case of synchronizing road coloring: the directed strongly connected graph of uniform outdegree has k-synchronizing coloring if and only if the great common divisor of length of its cycles is k. The corollary for arbitrary directed finite graph of uniform outdegree is also presented.