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 graphG $G$ moves infinitely many vertices, then there is a distinguishing 2‐colouring. We confirm this conjecture for graphs with maximum degreeΔ ≤ 5 ${\rm{\Delta }}\le 5$ . Furthermore, using similar techniques we show that if an infinite graph has maximum degreeΔ ≥ 3 ${\rm{\Delta }}\ge 3$ , then it admits a distinguishing colouring withΔ − 1 ${\rm{\Delta }}-1$ colours. This bound is sharp.