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 connected graph (Formula presented.) moves infinitely many vertices, then there is a distinguishing 2-colouring. We confirm this conjecture for graphs with maximum degree (Formula presented.). Furthermore, using similar techniques we show that if an infinite graph has maximum degree (Formula presented.), then it admits a distinguishing colouring with (Formula presented.) colours. This bound is sharp.