2-Colorability of r-Uniform Hypergraphs
A hypergraph is properly 2-colorable if each vertex can be colored by one of two colors and no edge is completely colored by a single color. We present a complete algebraic characterization of the 2-colorability of r-uniform hypergraphs. This generalizes a well known algebraic characterization of k-colorability of graphs due to Alon, Tarsi, Lovasz, de Loera, and Hillar. We also introduce a method for distinguishing proper 2-colorings called coloring schemes, and provide a decomposition of all proper 2-colorings into these schemes.
As an application, we present a new example of a 4-uniform non-2-colorable hypergraph on 11 vertices and 24 edges which is not isomorphic to a well-known construction by Seymour (1974) of a minimal non-2-colorable 4-uniform hypergraph. Additionally, we provide a heuristically constructed hypergraph which admits only specific coloring schemes. Further, we give an algebraic characterization of the coloring scheme known as a conflict-free coloring.