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

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...

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Published inJournal of algebra Vol. 607; pp. 64 - 106
Main Author Babai, László
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.10.2022
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ISSN0021-8693
1090-266X
DOI10.1016/j.jalgebra.2021.10.033

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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 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.
AbstractList 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.
Author Babai, László
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Keywords Profinite groups
Locally finite graphs
Asymmetry
Permutation groups
Minimal degree
Graphs
Setwise stabilizers
Language English
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Snippet 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...
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elsevier
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Publisher
StartPage 64
SubjectTerms Asymmetry
Graphs
Locally finite graphs
Minimal degree
Permutation groups
Profinite groups
Setwise stabilizers
Title Asymmetric coloring of locally finite graphs and profinite permutation groups: Tucker's Conjecture confirmed
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