On asymmetric colourings of graphs with bounded degrees and infinite motion

TL;DR

This paper advances the understanding of asymmetric graph colourings by showing that graphs with bounded degrees and infinite motion can be coloured asymmetrically with significantly fewer colours than previously known, specifically improving bounds for such graphs.

Contribution

The paper proves that graphs with bounded degree and infinite automorphism motion can be asymmetrically coloured with O(√Δ log Δ) colours, improving upon the trivial O(Δ) bound.

Findings

Established an upper bound of O(√Δ log Δ) colours for asymmetric colourings.

Progressed towards Tucker's conjecture for graphs with bounded degrees.

First known improvement over the trivial colour bound for this class of graphs.

Abstract

A vertex colouring of a graph is called asymmetric if the only automorphism which preserves it is the identity. Tucker conjectured that if every automorphism of a connected, locally finite graph moves infinitely many vertices, then there is an asymmetric colouring with $2$ colours. We make progress on this conjecture in the special case of graphs with bounded maximal degree. More precisely, we prove that if every automorphism of a connected graph with maximal degree $\Delta$ moves infinitely many vertices, then there is an asymmetric colouring using $\mathcal O(\sqrt \Delta \log \Delta)$ colours. This is the first improvement over the trivial bound of $\mathcal O(\Delta)$.