Graph Homomorphisms and Long Range Action

We show that if a graph H is k-colorable, then (k-1)-branching walks on H exhibit long range action, in the sense that the position of a token at time 0 constrains the configuration of its descendents arbitrarily far into the future. This long range action property is one of several investigated herein; all are similar in some respects to chromatic number but based on viewing H as the range, instead of the domain, of a graph homomorphism. The properties are based on combinatorial forms of probabilistic concepts from statistical physics, although we argue that they are natural even in a purely graph-theoretic setting. They behave well in many respects, but quite a few fundamental questions remain open.