University of Isfahan Transactions on Combinatorics 2251-8657 6 2 2017 06 01 Adjacent vertex distinguishing acyclic edge coloring of the Cartesian product of graphs 19 30 20988 10.22108/toc.2017.20988 EN Fatemeh Sadat Mousavi University of Zanjan Massomeh Noori University of Zanjan Journal Article 2016 01 13 ‎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 $6$ fo every $m\geq 3$ and $n\geq 3$‎. ‎Moreover in some cases we find the exact value of $\chi^{\prime}_{aa}(C_m\square C_n)$‎. https://toc.ui.ac.ir/article_20988_dc6050dc4f36dd95fd12e657ff895814.pdf