Vertex-coloring edge-weighting of complete \(r\)-partite graphs (Q2870955)
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scientific article; zbMATH DE number 6248708
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Vertex-coloring edge-weighting of complete \(r\)-partite graphs |
scientific article; zbMATH DE number 6248708 |
Statements
21 January 2014
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edge-weighting
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vertex-coloring
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complete \(r\)-partite graph
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1-2-3 conjecture
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Vertex-coloring edge-weighting of complete \(r\)-partite graphs (English)
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The authors provide a partial solution to the well-known 1-2-3 conjecture of \textit{M. KaroĊski} et al. [J. Comb. Theory, Ser. B 91, No. 1, 151--157 (2004; Zbl 1042.05045)]. Namely, they prove that in the case of a complete \(r\)-partite graph with \(n_i\) vertices in the \(i\)th partition sets, \(i=1,2,\dots, r\), the minimum number of colors needed to distinguish the vertex-weighted degrees is \(1\) when all \(n_i\) are distinct, \(3\) when \(n_1=n_2=\dots =n_r=1\), and \(2\) otherwise.
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