  We deal with $k$-out-regular directed multigraphs with loops (called simply
\emph{digraphs}). The edges of such a digraph can be colored by elements of
some fixed $k$-element set in such a way that outgoing edges of every vertex
have different colors. Such a coloring corresponds naturally to an automaton.
The road coloring theorem states that every primitive digraph has a
synchronizing coloring.
  In the present paper we study how many synchronizing colorings can exist for
a digraph with $n$ vertices. We performed an extensive experimental
investigation of digraphs with small number of vertices. This was done by using
our dedicated algorithm exhaustively enumerating all small digraphs. We also
present a series of digraphs whose fraction of synchronizing colorings is equal
to $1-1/k^d$, for every $d \ge 1$ and the number of vertices large enough.
  On the basis of our results we state several conjectures and open problems.
In particular, we conjecture that $1-1/k$ is the smallest possible fraction of
synchronizing colorings, except for a single exceptional example on 6 vertices
for $k=2$.
