Distinguishing infinite graphs with bounded degrees
Florian Lehner, Monika Pilśniak, Marcin Stawiski · Journal of Graph Theory · 2022
Abstract 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 moves infinitely many vertices, then there is a distinguishing 2‐colouring. We confirm this conjecture for graphs with maximum degree . Furthermore, using similar techniques we show that if an infinite graph has maximum degree , then it admits a distinguishing colouring with colours. This bound is sharp.