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

An asymmetric coloring of a graph is its vertices that not preserved by any non-identity automorphism the graph. The motion minimal degree group, i.e., minimum number elements are moved (not fixed) automorphism. We confirm Tom Tucker's “Infinite Motion Conjecture” connected locally finite graphs with infinite admit an 2-coloring. infer this from more general result inverse limit sequence permutation groups disjoint domains, viewed as group on union those admits proof based study interaction between epimorphisms and structure setwise stabilizers subsets their domains. note connections subject to computational theory, asymptotic highly regular structures, Graph Isomorphism problem, list open problems.