The complexity of L(p,q)-Edge-Labelling

We consider the L(p,q)-Edge-Labelling problem, which is the edge variant of the well-known L(p,q)-Labelling problem. So far, the complexity of this problem was only partially classified. We complete this study for all nonnegative p and q, by showing that, whenever (p,q) is not (0,0), L(p,q)-Edge-Labelling problem is NP-complete. We do this by proving that for all nonnegative p and q, except p=q=0, there exists an integer k so that L(p,q)-Edge-k-Labelling is NP-complete.