  Fix a simple graph $G=(V,E)$ and choose a random initial 3-coloring of
vertices drawn from a uniform product measure. The 3-color cycle cellular
automaton is a process in which at each discrete time step in parallel, every
vertex with color $i$ advances to the successor color $(i+1)$ mod 3 if in
contact with a neighbor with the successor color, and otherwise retains the
same color. In the Greenberg-Hastings Model, the same update rule applies only
to color 0, while other two colors automatically advance. The limiting behavior
of these processes has been studied mainly on the integer lattices. In this
paper, we introduce a monotone comparison process defined on the universal
covering space of the underlying graph, and characterize the limiting behavior
of these processes on arbitrary connected graphs. In particular, we establish a
phase transition on the Erd\"os-R\'enyi random graph. On infinite trees, we
connect the rate of color change to the cloud speed of an associated
tree-indexed walk. We give estimates of the cloud speed by generalizing known
results to trees with leaves.
