Number of colors needed to break symmetries of a graph by an arbitrary edge coloring

A coloring is distinguishing (or symmetry breaking) if no non-identity automorphism preserves it. The threshold of a graph G, denoted by θ(G), the minimum number colors k so that every k-coloring G distinguishing. We generalize this concept to edge-coloring defining an alternative index θ′(G). consider θ′ for some families graphs and find its connection with edge-cycles group. Then we show θ′(G) = 2 only ≃ K1, 3 P4, or K3. Moreover, prove auxiliary results whose although there are infinitely many such graphs, but they not line graphs. Finally, compute when Cartesian product simple prime