Centre for Discrete and Applicable Mathematics

 CDAM Research Report, LSE-CDAM-2001-07

September 2001

Graph Homomorphisms and Long Range Action

Graham R. Brightwell and Peter Winkler


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.

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