Arbitrarily large difference between d-strong chromatic index and its trivial lower bound

We present some general results on d-strong edge colorings of circulant graphs. We determine exact values of the d-strong chromatic index for circulant graphs Cn(1,2) for d=1 and d=2, we also prove that the difference between the d-strong chromatic index and its trivial lower bound can be arbitrarily large.