Local finiteness, distinguishing numbers and Tucker's conjecture
  A distinguishing colouring of a graph is a colouring of the vertex set such that no non-trivial automorphism preserves the colouring. Tucker conjectured that if every non-trivial automorphism of a locally finite graph moves infinitely many vertices, then there is a distinguishing 2-colouring.   We show that the requirement of local finiteness is necessary by giving a non-locally finite graph for which no finite number of colours suffices. 
Florian Lehner and R\"ognvaldur G. M\"oller
math.CO