Distinguishing infinite graphs with bounded degrees

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 $\mathrm{\Delta }\le 5$. Furthermore, using similar techniques we show that if an infinite graph has maximum degree $\mathrm{\Delta }\ge 3$, then it admits a distinguishing colouring with $\mathrm{\Delta }-1$ colours. This bound is sharp.