On asymmetric colourings of graphs with bounded degrees and infinite motion

A vertex colouring of a graph is called asymmetric if the only automorphism which preserves it is the identity. Tucker conjectured that if every automorphism of a connected, locally finite graph moves infinitely many vertices, then there is an asymmetric colouring with $2$ colours. We make progress on this conjecture in the special case of graphs with bounded maximal degree. More precisely, we prove that if every automorphism of a connected graph with maximal degree $\Delta$ moves infinitely many vertices, then there is an asymmetric colouring using $\mathcal O(\sqrt \Delta \log \Delta)$ colours. This is the first improvement over the trivial bound of $\mathcal O(\Delta)$.