Distinguishing graphs with intermediate growth

AbstractA graph G is said to be 2-distinguishable if there is a 2-labeling of its vertices which is not preserved by any nontrivial automorphism of G. We show that every locally finte graph with infinite nite motion and growth at most $$\mathcal{O}\left( {2^{(1 - \varepsilon )\tfrac{{\sqrt n }} {2}} } \right)$$ is 2-distinguishable. Infinite motion means that every automorphism moves infinitely many vertices and growth refers to the cardinality of balls of radius n.

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