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

TL;DR

This paper proves Tucker's Infinite Motion Conjecture, showing that connected locally finite graphs with infinite automorphism motion can be asymmetrically 2-colored, using inverse limits of finite permutation groups.

Contribution

It confirms Tucker's conjecture by establishing that such graphs admit an asymmetric 2-coloring, extending the result to inverse limits of finite permutation groups.

Findings

Connected locally finite graphs with infinite motion admit asymmetric 2-colorings.

The proof involves inverse limits of finite permutation groups and their stabilizers.

The approach links epimorphisms of permutation groups to graph automorphism properties.

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.