  Given a finite directed graph, a coloring of its edges turns the graph into a
finite-state automaton. A k-synchronizing word of a deterministic automaton is
a word in the alphabet of colors at its edges that maps the state set of the
automaton at least on k-element subset. A coloring of edges of a directed
strongly connected finite graph of a uniform outdegree (constant outdegree of
any vertex) is k-synchronizing if the coloring turns the graph into a
deterministic finite automaton possessing a k-synchronizing word.
  For k=1 one has the well known road coloring problem. The recent positive
solution of the road coloring problem implies an elegant generalization
considered first by Beal and Perrin: a directed finite strongly connected graph
of uniform outdegree is k-synchronizing iff the greatest common divisor of
lengths of all its cycles is k.
  Some consequences for coloring of an arbitrary finite digraph are presented.
  We describe a subquadratic algorithm of the road coloring for the
k-synchronization implemented in the package TESTAS. A new linear visualization
program demonstrates the obtained coloring. Some consequences for coloring of
an arbitrary finite digraph and of such a graph of uniform outdegree are
presented.
