A uniform hypergraph is properly k-colorable if each vertex is colored by one of k colors and no edge is completely colored by one color. In 2008 Hillar gave a complete characterization of the k-colorability of graphs through algebraic methods. We generalize Hillar's work and give a complete algebraic characterization of the k-colorability of r−uniform hypergraphs. In addition to general k colorability, we provide an alternate characterization for 2-colorability and apply this to some constructions of hypergraphs that are minimally non-2-colorable.;We also provide examples and verification of minimality for non-2-colorable 5- and 6-uniform hypergraphs. As a further application, we give a characterization for a uniform hypergraph to be conflict-free colorable.;Finally, we provide an improvement on the construction introduced by Abbott and Hanson in 1969, and improved upon by Seymour in 1974.
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机译：如果每个顶点都用 k 颜色之一着色并且没有边缘完全用一种颜色着色，则统一的超图就可以正确地 k 着色。 Hillar在2008年通过代数方法对图的 k 可着色性进行了完整的表征。我们对Hillar的工作进行了概括，并对 r -统一超图的 k -可着色性进行了完整的代数表征。除了一般的 k 着色性之外，我们还提供了2着色性的另一种表征，并将其应用于至少不可2着色的超图的某些构造。非2色5和6均匀超图。作为进一步的应用，我们给出了一个统一的超图的特征，使其无冲突可着色。最后，我们对1969年由雅培（Abbott）和汉森（Hanson）提出并在1974年由西摩（Seymour）进行了改进。
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