Asymmetrizing infinite trees

TL;DR

This paper extends the understanding of asymmetrizing infinite trees by demonstrating that for any infinite motion, such trees can be asymmetrized, and the number of inequivalent asymmetrizing sets is maximally large.

Contribution

It generalizes previous results by showing that infinite trees with arbitrary infinite motion are asymmetrizable, and quantifies the vast number of asymmetrizing sets.

Findings

Infinite trees with any infinite motion are asymmetrizable.

Number of inequivalent asymmetrizing sets is 2^{|T|}.

Generalizes known results for bounded degree trees.

Abstract

A graph $G$ is asymmetrizable if it has a set of vertices whose setwise stablizer only consists of the identity automorphism. The motion $m$ of a graph is the minimum number of vertices moved by any non-identity automorphism. It is known that infinite trees $T$ with motion $m=\aleph_0$ are asymmetrizable if the vertex-degrees are bounded by $2^m.$ We show that this also holds for arbitrary, infinite $m$, and that the number of inequivalent asymmetrizing sets is $2^{|T|}$.