Breaking graph symmetries by edge colourings

TL;DR

This paper proves a conjecture that for countable graphs where every non-trivial automorphism moves infinitely many edges, only two edge colours are needed to break all symmetries.

Contribution

It confirms Broere and Pilśniak's conjecture, establishing that such graphs have a distinguishing index of at most two.

Findings

Confirmed the conjecture for countable graphs

Established that infinite edge movement implies D'(G) ≤ 2

Advances understanding of graph symmetry breaking

Abstract

The distinguishing index $D'(G)$ of a graph $G$ is the least number of colours needed in an edge colouring which is not preserved by any non-trivial automorphism. Broere and Pil\'sniak conjectured that if every non-trivial automorphism of a countable graph $G$ moves infinitely many edges, then $D'(G) \leq 2$. We prove this conjecture.