Distinguishing infinite graphs with bounded degrees

TL;DR

This paper proves Tucker's conjecture for infinite graphs with maximum degree up to 5, and shows such graphs with degree at least 3 can be distinguished with fewer colours, improving understanding of graph symmetries.

Contribution

It confirms Tucker's conjecture for graphs with maximum degree ≤ 5 and establishes a sharp bound for distinguishing colourings with Δ - 1 colours for higher degrees.

Findings

Confirmed Tucker's conjecture for Δ ≤ 5.

Established Δ - 1 colour bound for graphs with Δ ≥ 3.

Proved the bound is sharp.

Abstract

Call a colouring of a graph distinguishing, if the only colour preserving automorphism is the identity. A conjecture of Tucker states that if every automorphism of a graph $G$ moves infinitely many vertices, then there is a distinguishing $2$-colouring. We confirm this conjecture for graphs with maximum degree $\Delta \leq 5$. Furthermore, using similar techniques we show that if an infinite graph has maximum degree $\Delta \geq 3$, then it admits a distinguishing colouring with $\Delta - 1$ colours. This bound is sharp.