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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
##### Keywords:
total weight choosable
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##### References:
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