The L(p, q)-Edge-Labelling problem is the edge variant of the well-known L(p, q)-Labelling problem. It is equivalent to the L(p, q)-Labelling problem itself if we restrict the input of the latter problem to line graphs. So far, the complexity of L(p, q)-Edge-Labelling was only partially classified in the literature. We complete this study for all p, q ≥ 0 by showing that whenever (p, q) = (0, 0), the L(p, q)-Edge-Labelling problem is NP-complete. We do this by proving that for all p, q ≥ 0 except p = q = 0, there is an integer k so that L(p, q)-Edge-k-Labelling is NP-complete.

The L(p, q)-Edge-Labelling problem is the edge variant of the well-known L(p, q)-Labelling problem. It is equivalent to the L(p, q)-Labelling problem itself if we restrict the input of the latter problem to line graphs. So far, the complexity of L(p, q)-Edge-Labelling was only partially classified in the literature. We complete this study for all p, q ≥ 0 by showing that whenever (p, q) = (0, 0), the L(p, q)-Edge-Labelling problem is NP-complete. We do this by proving that for all p, q ≥ 0 except p = q = 0, there is an integer k so that L(p, q)-Edge-k-Labelling is NP-complete.

We classify the complexity of L(p, q)-Edge-k-Labelling in the sense that for all positive integers p and q we exhibit k so that we can show L(p, q)-Edge-k-Labelling is NP-complete.

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