A $k$-edge-weighting of $G$ is a mapping $\omega:E(G)\longrightarrow \{1,\ldots,k\}$. The edge-weighting of $G$ naturally induces a vertex-colouring $\sigma_{\omega}:V(G)\longrightarrow \mathbb{N}$ given by$\sigma_{\omega}(v)=\sum_{u\in N_G(v)}\omega(vu)$ for every $v\in V(G)$. The edge-weighting $\omega$ is neighbour sum distinguishing if it yields a proper vertex-colouring $\sigma_{\omega}$, \emph{i.e.}, $\sigma_{\omega}(u)\neq \sigma_{\omega}(v)$ 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. A 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 $(\Delta(G)+2)$-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.

翻译：$G$的一个$k$-边赋权是指映射$\omega:E(G)\longrightarrow \{1,\ldots,k\}$。该边赋权自然诱导出一个由$\sigma_{\omega}(v)=\sum_{u\in N_G(v)}\omega(vu)$（$\forall v\in V(G)$）定义的顶点着色$\sigma_{\omega}:V(G)\longrightarrow \mathbb{N}$。若边赋权$\omega$能产生正常顶点着色$\sigma_{\omega}$，即对$G$的每条边$uv$均有$\sigma_{\omega}(u)\neq \sigma_{\omega}(v)$，则称其为邻和可区别的。本文研究具有局部约束的邻和可区别边赋权，具体而言，我们假设大度顶点关联的边集并非单色的。若图不含同构于$K_2$的分支，则称其为优良图。我们证明每个最大度不超过5的优良图均存在一个邻和可区别的$(\Delta(G)+2)$-边赋权，使得所有度数至少为2的顶点都至少关联两条不同权重的边。进一步地，我们证明每个优良图均存在一个邻和可区别的$7$-边赋权，使得所有度数至少为6的顶点都至少关联两条不同权重的边。最后，我们证明优良二部图存在一个邻和可区别的$6$-边赋权，使得所有度数至少为2的顶点都至少关联两条不同权重的边。