A star edge coloring of a graph G is a proper edge coloring where at least three distinct colors are used on the edges of every path and cycle of length four.The star chromatic index of G is the minimum number of colors in a star edge coloring of G. In this theses we present upper bounds for the star chromatic index of cartesian products of some graphs. we give an sharp upper bound for hypercube graphs. Also we find the exact value of the strong chromatic index for cartesian product of two paths.
