The Road Coloring for Mapping on k States(withdrawn)
  Let $\Gamma$ be directed strongly connected finite graph of uniform outdegree (constant outdegree of any vertex) and let some coloring of edges of $\Gamma$ 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 $\Gamma$ and let $\Gamma s$ be a mapping of vertices $\Gamma$. 
A. N. Trahtman