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An asymmetric coloring of a graph is a coloring of its vertices that is notpreserved by any non-identity automorphism of the graph. The motion of a graphis the minimal degree of its automorphism group, i.e., the minimum number ofelements displaced by any non-identity automorphism. In this paper we confirmTom Tucker's "Infinite Motion Conjecture" that connected locally finite graphswith infinite motion admit an asymmetric 2-coloring. We infer this from themore general result that the inverse limit of a sequence of finite permutationgroups with disjoint domains, viewed as a permutation group on the union ofthose domains, admits an asymmetric 2-coloring. The proof is based on the studyof the interaction between epimorphisms of finite permutation groups and thestructure of the setwise stabilizers of subsets of their domains.

Asymmetric coloring of locally finite graphs and profinite permutation groups: Tucker's Conjecture confirmed