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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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