adjacent vertex distinguishing acyclic edge coloring of the cartesian product of graphs

‎let g be a graph and χ′aa(g) denotes the minimum number of colors required for an‎ ‎acyclic edge coloring of g in which no two adjacent vertices are incident to edges colored with the same set of colors‎. ‎we prove a general bound for χ′aa(g□h) for any two graphs g and h‎. ‎we also determine‎ ‎exact value of this parameter for the cartesian product of two paths‎, ‎cartesian product of a path and a cycle‎, ‎cartesian product of two trees‎, ‎hypercubes‎. ‎we show that χ′aa(cm□cn) is at most 6 fo every m≥3 and n≥3‎. ‎moreover in some cases we find the exact value of χ′aa(cm□cn)‎.