Adjacent vertex distinguishing acyclic edge coloring of the Cartesian product of graphs

‎Let G be a graph and \chi^{\prime}_{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 \chi^{\prime}_{aa}(G\square 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 \chi^{\prime}_{aa}(C_m\square C_n) is at most ۶ fo every m\geq ۳ and n\geq ۳‎. ‎Moreover in some cases we find the exact value of \chi^{\prime}_{aa}(C_m\square C_n)‎.

Keywords:

‎Acyclic edge coloring‎ , ‎adjacent vertex distinguishing acyclic edge coloring‎ , ‎adjacent vertex distinguishing acyclic edge chromatic number