Phase transition in firefly cellular automata on finite trees

We study a one-parameter family of discrete dynamical systems called the κ-color firefly cellular automata, proposed by the author in [13] as discrete models for finite-state pulse-coupled inhibitory oscillators. At each discrete time t , each vertex in a graph has a state (color) in {0, · · · ,κ−1}, and a special color b(κ) = bκ−1 2 c is designated as the ‘blinking’ color. During each transition from time t to t +1, each vertex simultaneously increment its color from i to i +1 moduloκ unless i > b(κ) and at least one of its neighbors is at the blinking color b(κ), in which case it stays at the same color. We say a given initial κ-coloring on vertices of a graph synchronizes if it evolves to a constant coloring so that all vertices increment their color by 1 in unison thereafter. It is easy to see that if every initial κ-coloring on a fixed finite tree synchronizes, then necessarily the tree has maximum degree strictly less than κ. We show that the converse of this statement is true for κ ∈ {3,4,5,6}, but we also provide counterexamples for κ= 7 up to 30, exhibiting a phase transition in our model on finite trees “between” κ= 6 and κ= 7.