Precise bounds for the distinguishing index of the Cartesian product

The distinguishing index D'(G) of a graph G is the least number d such that G has an edge colouring with d colours that is preserved only by the identity automorphism. The distinguishing index of the Cartesian product of graphs was investigated by the authors and Kalinowski. They considered colourings with two colours only and obtained results that do not determine the distinguishing index for all the possible cases. In this paper we investigate colourings with d colours and determine the exact value of the distinguishing index of the Cartesian product K-1,K-m square K-1,K-h for almost all m and n. In particular, we supplement the result of [6] for the case when 2(2m+1) - [ m/2 ] +1 < n < 2(2m+1). We also observe the distinguishing index of the Cartesian product of two graphs in general does not have to depend on the size of the graphs and it can be arbitrarily small. (C) 2017 Elsevier B.V. All rights reserved.