Bounds for Distinguishing Invariants of Infinite Graphs

We consider infinite graphs. The distinguishing number D(G) of a graph G is the minimum number of colours in a vertex colouring of G that is preserved only by the trivial automorphism. An analogous invariant for edge colourings is called the distinguishing index, denoted by D′(G). We prove that D′(G) 6 D(G) + 1. For proper colourings, we study relevant invariants called the distinguishing chromatic number χD(G), and the distinguishing chromatic index χ ′ D(G), for vertex and edge colourings, respectively. We show that χD(G) 6 2∆(G)− 1 for graphs with a finite maximum degree ∆(G), and we obtain substantially lower bounds for some classes of graphs with infinite motion. We also show that χD(G) 6 χ ′(G) + 1, where χ′(G) is the chromatic index of G, and we prove a similar result χ′′ D(G) 6 χ ′′(G) + 1 for proper total colourings. A number of conjectures are formulated.