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Neighbor sum distinguishing index. (English) Zbl 1272.05047
Summary: We consider proper edge colorings of a graph $$G$$ using colors of the set $$\{1,\dots ,k\}$$. Such a coloring is called neighbor sum distinguishing if for any pair of adjacent vertices $$x$$ and $$y$$ the sum of colors taken on the edges incident to $$x$$ is different from the sum of colors taken on the edges incident to $$y$$. The smallest value of $$k$$ in such a coloring of $$G$$ is denoted by $$\mathrm{ndi}_\Sigma (G)$$. In the paper we conjecture that for any connected graph $$G\neq C_5$$ of order $$n\geq 3$$ we have $$\mathrm{ndi}_\Sigma (G)\leq\Delta (G)+2$$. We prove this conjecture for several classes of graphs. We also show that $$\mathrm{ndi}_\Sigma (G)\leq 7\Delta (G)/2$$ for any graph $$G$$ with $$\Delta (G)\geq 2$$ and $$\mathrm{ndi}_\Sigma (G)\leq 8$$ if $$G$$ is cubic.

##### MSC:
 05C15 Coloring of graphs and hypergraphs
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