  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.
