No abstract

The distinguishing index D (G) of a graph G is the least natural number d such that G has an edge colouring with d colours that is only preserved by the identity automorphism. In this paper we investigate the distinguishing index of the Cartesian product of connected finite graphs. We prove that for every k ≥ 2, the k-th Cartesian power of a connected graph G has distinguishing index equal 2, with the only exception D (K 2 2) = 3. We also prove that if G and H are connected graphs that satisfy the relation 2 ≤ |G| ≤ |H| ≤ 2 |G| 2 G − 1 − |G| + 2, then D (G2H) ≤ 2 unless G2H = K 2 2 .

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) D(G) + 1. For proper colourings, we study relevant invariants called the distinguishing chromatic number χ D (G), and the distinguishing chromatic index χ D (G), for vertex and edge colourings, respectively. We show that χ D (G) 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 χ D (G) χ (G) + 1, where χ (G) is the chromatic index of G, and we prove a similar result χ D (G) χ (G) + 1 for proper total colourings. A number of conjectures are formulated.

A proper edge-coloring of a graph defines at each vertex the set of colors of its incident edges. This set is called the palette of the vertex. In this paper we are interested in the minimum number of palettes taken over all possible proper colorings of a graph.

The distinguishing number D(G) of a graph G is the minimum number of colors needed to color the vertices of G such that the coloring is preserved only by the trivial automorphism. In this paper we improve results about the distinguishing number of Cartesian products of finite and infinite graphs by removing restrictions to prime or relatively prime factors.