Coarse distinguishability of graphs with symmetric growth

TL;DR

This paper demonstrates that graphs with symmetric growth can be asymmetrically colored to limit automorphisms and verifies the infinite motion conjecture for graphs with certain stabilizer properties, advancing understanding of graph symmetries.

Contribution

It introduces a coarse symmetry-breaking coloring for graphs with symmetric growth and proves the infinite motion conjecture under specific stabilizer conditions.

Findings

Existence of a vertex coloring that constrains automorphisms to be close to identity.

Validation of the infinite motion conjecture for graphs with unbounded stabilizer actions.

Establishment of a coarse geometric approach to symmetry breaking in graphs.

Abstract

Let $X$ be a connected, locally finite graph with symmetric growth. We prove that there is a vertex coloring $\phi\colon X\to\{0,1\}$ and some $R\in\mathbb{N}$ such that every automorphism $f$ preserving $\phi$ is $R$-close to the identity map; this can be seen as a coarse geometric version of symmetry breaking. We also prove that the infinite motion conjecture is true for graphs where at least one vertex stabilizer $S_x$ satisfies the following condition: for every non-identity automorphism $f\in S_x$, there is a sequence $x_n$ such that $\lim d(x_n,f(x_n))=\infty$.