Breaking small automorphisms of graphs of arbitrary cardinality

TL;DR

This paper proves that for any connected graph, finite or infinite, it is possible to colour edges with two colours to break all small automorphisms, extending previous results from finite graphs.

Contribution

The paper confirms the conjecture that two colours suffice to break all small automorphisms in connected graphs of any size, finite or infinite.

Findings

Two-colour edge colouring breaks all small automorphisms in connected graphs.

The result extends to graphs of arbitrary cardinality, including infinite graphs.

Confirms a conjecture for finite graphs and generalizes it to infinite cases.

Abstract

We say that an edge colouring $c$ of a graph preserves an automorphism $\varphi$ if $\varphi$ maps each edge to an edge of the same colour. Otherwise, we say that $c$ breaks $\varphi$. We call an automorphism of a graph small if it moves some vertex to its neighbour. We study the edge colourings of graphs that break every small automorphism. Kalinowski, Pil\'sniak, and Wo\'zniak proved that three colours are enough for such a colouring to exist for every finite graph without isolated edges. They conjectured that two colours are enough for every finite connected graph on at least six vertices. We confirm this conjecture in its more general version, namely for connected finite and infinite graphs of arbitrary cardinality.