Hypergraph Colorings, Commutative Algebra, and Gröbner Bases

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 and Windfeldt gave a complete characterization of the k-colorability of graphs through algebraic methods. We generalize their 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