ADJACENT VERTEX DISTINGUISHING ACYCLIC EDGE COLORING OF THE CARTESIAN PRODUCT OF GRAPHS

Let G be a graph and xaa ′ 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 xaa ′ 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 xaa ′ aa(Cm□Cn) is at most 6 fo every m 3 and n 3. Moreover in some cases we nd the exact value of xaa ′ aa(Cm□Cn).