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

• Published in: Journal of algebra Vol. 607; pp. 64 - 106
• Main Author: Babai, László
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.10.2022
• Subjects: Asymmetry; Graphs; Locally finite graphs; Minimal degree; Permutation groups; Profinite groups; Setwise stabilizers; Profinite groups; Locally finite graphs; Asymmetry; Permutation groups; Minimal degree; Graphs; Setwise stabilizers
• ISSN: 0021-8693; 1090-266X
• DOI: 10.1016/j.jalgebra.2021.10.033

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