Distinguishing infinite star-free graphs

Abstract

Call a vertex or an edge colouring of a graph distinguishing, if is not preserved by any non-identity automorphism. For a graph H, we say that a graph G is H-free if there is no induced subgraph of G, which is isomorphic to H. Gorzkowska, Kargul, Musiał and Pal proved that for every natural number n greater than 2 each finite connected $K_{1,n}$-free graph on at least six vertices has a distinguishing edge colouring using at most $n-1$ colours. We extend this result to all locally finite connected $K_{1,n}$-free graphs on at least six vertices.