Date of Award
Doctor of Philosophy in Mathematics
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.
Krul, Michael E. M., "Hypergraph Colorings, Commutative Algebra, and Gröbner Bases" (2013). Open Access Dissertations. Paper 20.