The neighbour sum distinguishing edge-weighting with local constraints
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A k-edge-weighting of G is a mapping ω:E(G)⟶{1,…,k}. The edge-weighting naturally induces a vertex colouring σ_ω:V(G)⟶ℕ given by σ_ω(v)=∑_u∈ N_G(v)ω(vu) for every v∈ V(G). The edge-weighting ω is neighbour sum distinguishing if it yields a proper vertex colouring σ_ω, i.e., σ_ω(u)≠σ_ω(v) for every edge uv of G.We investigate a neighbour sum distinguishing edge-weighting with local constraints, namely, we assume that the set of edges incident to a vertex of large degree is not monochromatic. The graph is nice if it has no components isomorphic to K_2. We prove that every nice graph with maximum degree at most 5 admits a neighbour sum distinguishing (Δ(G)+2)-edge-weighting such that all the vertices of degree at least 2 are incident with at least two edges of different weights. Furthermore, we prove that every nice graph admits a neighbour sum distinguishing 7-edge-weighting such that all the vertices of degree at least 6 are incident with at least two edges of different weights. Finally, we show that nice bipartite graphs admit a neighbour sum distinguishing 6-edge-weighting such that all the vertices of degree at least 2 are incident with at least two edges of different weights.
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