Abstract
The distinguishing index D ′ ( G ) of a graph G is the least cardinal d such that G has an edge colouring with d colours that is preserved only by the trivial automorphism. We derive some bounds for this parameter for infinite graphs. In particular, we investigate the distinguishing index of the Cartesian product of countable graphs. Finally, we prove that D ʹ( K 2 ℵ 0 ) = 2 , where K 2 ℵ 0 is the infinite dimensional hypercube.
Highlights
Albertson and Collins [1] introduced thedistinguishing number D(G) of a graph G as the least cardinal d such that G has a labelling with d labels that is only preserved by the trivial automorphism
Countable infinite graphs have been investigated with respect to the distinguishing number; see [12], [13], and [14]
The distinguishing index D (G) of a graph G is the least cardinal d such that G has an edge colouring with d colours that is only preserved by the trivial automorphism
Summary
Albertson and Collins [1] introduced the (vertex-)distinguishing number D(G) of a graph G as the least cardinal d such that G has a labelling with d labels that is only preserved by the trivial automorphism. A basic fact, which is a reformulation of the above theorem for r = 2 and which is used frequently in this paper, is: If φ is an automorphism of the Cartesian product G12G2 of two connected relatively prime graphs, there are automorphisms φi of Gi, i = 1, 2, such that φ(v1, v2) = (φ1(v1), φ2(v2)) for all (v1, v2) ∈ V (G12G2). The distinguishing index of the Cartesian product of finite graphs is investigated in [4] where the authors prove, amongst others, a result which will be useful and which we record as Theorem 1.4.
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