Asymmetric coloring of locally finite graphs and profinite permutation groups: Tucker's Conjecture confirmed
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Publication:6380510
DOI10.1016/J.JALGEBRA.2021.10.033arXiv2110.08492WikidataQ123207189 ScholiaQ123207189MaRDI QIDQ6380510
Publication date: 16 October 2021
Abstract: 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 displaced by any non-identity automorphism. In this paper 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 a 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.
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