Automorphisms of graphs and a characterization of lattices

An automorphism $g$ of an undirected connected graph $\Gamma$ is called bounded if for some natural number $c$and an arbitrary vertex $\alpha$ of the graph $\Gamma$ the inequality $d(\alpha,g(\alpha))<c$.
The structure of vertex-transitive groups of bounded automorphisms of locally finite graphs is studied. A characterization of locally finite graphs which admit a vertex-transitive group of bounded automorphisms is obtained.
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