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{\bf Jakub Przyby{\l}o and Mariusz Wo\'zniak}
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{\bf Total Weight Choosability of Graphs}
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Suppose the edges and the vertices of a simple graph $G$ are assigned
$k$-element lists of real weights. By choosing a representative of
each list, we specify a vertex colouring, where for each vertex its
colour is defined as the sum of the weights of its incident edges and
the weight of the vertex itself. How long lists ensures a choice
implying a proper vertex colouring for any graph? Is there any finite
bound or maybe already lists of length two are sufficient? We prove
that $2$-element lists are enough for trees, wheels, unicyclic and
complete graphs, while the ones of length $3$ are sufficient for
complete bipartite graphs. Our main tool is an algebraic theorem by
Alon called Combinatorial Nullstellensatz.
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