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 aný ýacyclic edge coloring of G in which no two adjacent vertices are incident to edges colored with the same set of colorsý. ýWe prove a general bound for χ′aa(G□H) for any two graphs G and Hý. ýWe also determineý ýexact value of this parameter for the Cartesian product of two pathsý, ýCartesian product of a path and a cycleý, ýCartesian product of two treesý, ýhypercubesý. ýWe show that χ′aa(Cm□Cn) is at most 6 fo every m≥3 and n≥3ý. ýMoreover in some cases we find the exact value of χ′aa(Cm□Cn)ý.