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A paintability version of the combinatorial Nullstellensatz, and list colorings of \(k\)-partite \(k\)-uniform hypergraphs. (English) Zbl 1201.05039
Summary: We study the list coloring number of \(k\)-uniform \(k\)-partite hypergraphs. Answering a question of R. Ramamurthi and D.B. West [“Hypergraph extension of the Alon-Tarsi list coloring theorem,” Combinatorica 25, No. 3, 355–366 (2005; Zbl 1080.05034)], we present a new upper bound which generalizes N. Alon and M. Tarsi’s bound for bipartite graphs [“Colorings and orientations of graphs,” Combinatorica 12, No. 2, 125–134 (1992; Zbl 0756.05049)], the case \(k = 2\). Our results hold even for paintability (on-line list colorability). To prove this additional strengthening, we provide a new subject-specific version of the Combinatorial Nullstellensatz.

MSC:
05C15 Coloring of graphs and hypergraphs
11C08 Polynomials in number theory
91A43 Games involving graphs
05C65 Hypergraphs
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
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