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Total weight choosability of graphs. (English) Zbl 1228.05161
A graph \(G= (V, E)\) is called \((k, k')\)-total weight choosable if the following holds: For any mapping \(L\) which assigns to each vertex \(x\) a set \(L(x)\) of \(k\) real numbers, and assigns to each edge \(e\) a set \(L(e)\) of \(k'\) real numbers, there is a mapping \(f: V\cup E\to\mathbb{R}\) such that \(f(y)\in L(y)\) for any \(y\in V\cup E\) and for any two adjacent vertices \(x\), \(x'\), \[ \sum_{e\in E(x)} f(e)+ f(x)\neq \sum_{c\in E(x')} f(e)+ f(x'). \] Complete graphs, complete bipartite graphs and trees other than \(K_2\) are \((1,3)\)-total weight choosable. The authors conjecture that every graph is \((2,2)\)-total weight choosable and every graphs without isolated vertices is \((1,3)\)-total weight choosable. The authors prove that for any graph \(H\), a graph \(G\) obtained from \(H\) by subdividing each edge with at least two vertices is \((2, 2)\)-total weight choosable.

MSC:
05C15 Coloring of graphs and hypergraphs
05C78 Graph labelling (graceful graphs, bandwidth, etc.)
05C22 Signed and weighted graphs
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