Star edge coloring of the Cartesian product of graphs

The star chromatic index of a graph G is the smallest integer k for which G admits a proper edge coloring with k colors such that there is no bi-colored path nor cycle of length four. In this paper, we first obtain an upper bound for the star chromatic index of the Cartesian product of two graphs. We then determine the exact value of the star chromatic index of 2-dimensional grids. We also obtain upper bounds on the star chromatic index of the Cartesian product of a path with a cycle, d-dimensional grids, d-dimensional hypercubes and d-dimensional toroidal grids, for every natural number d >= 2.