Distinguishing graphs with intermediate growth

TL;DR

This paper proves that certain infinite, locally finite graphs with controlled growth and infinite automorphism motion can be distinguished with just two labels, advancing understanding of graph symmetries.

Contribution

It establishes a new bound on graph growth that guarantees 2-distinguishability for graphs with infinite automorphism motion.

Findings

Graphs with growth at most O(2^((1-ε)√n/2)) are 2-distinguishable.

Infinite motion ensures no automorphism fixes all vertices.

The result links growth rate and symmetry-breaking in graphs.

Abstract

A graph G is said to be 2-distinguishable if there is a 2-labeling of its vertices which is not preserved by any nontrivial automorphism of G. We show that every locally finite graph with infinite motion and growth at most O(2^((1-\varepsilon) \sqrt(n)/2)) is 2-distinguishable. Infinite motion means that every automorphism moves infinitely many vertices and growth refers to the cardinality of balls of radius n.