The distinguishing index of the Cartesian product of countable graphs

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.

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