  The road colouring theorem characterizes the class of strongly connected
directed graphs with constant out-degree that admit a synchronizing road
colouring. The subject of this paper is a pair of related conjectures that
generalize the road colouring theorem to graphs with non-constant out-degree;
we give reasons to believe that both of these conjectures are true. Our main
results focus on two classes of graphs, proving both conjectures for one class
of graphs and one of the conjectures for an additional class of graphs. We also
present computer simulations that give some empirical evidence for the
conjectures.
