We supply an upper bound on the distinguishing chromatic number of certain
infinite graphs satisfying an adjacency property. Distinguishing proper
n-colourings are generalized to the new notion of distinguishing
homomorphisms. We prove that if a graph G satisfies the connected
existentially closed property and admits a homomorphism to H, then it admits
continuum-many distinguishing homomorphisms from G to H join K2​.
Applications are given to a family universal H-colourable graphs, for H a
finite core