Every nice graph (no K2 components) with Δ≤5 admits a neighbour-sum-distinguishing (Δ+2)-edge-weighting where deg≥2 vertices have at least two distinct incident weights; every nice graph admits such a 7-weighting for deg≥6 vertices; nice bipartite graphs admit a 6-weighting for deg≥2 vertices.
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Neighbour sum distinguishing edge-weightings with local constraints
Every nice graph (no K2 components) with Δ≤5 admits a neighbour-sum-distinguishing (Δ+2)-edge-weighting where deg≥2 vertices have at least two distinct incident weights; every nice graph admits such a 7-weighting for deg≥6 vertices; nice bipartite graphs admit a 6-weighting for deg≥2 vertices.