Abstract
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.
Highlights
Colorability of graphs has a rich and extensive history and includes many different techniques
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
De Loera et al and Hillar proved results concerning the algebraic characterization of a graph colorability [7], [12]
Summary
Colorability of graphs has a rich and extensive history and includes many different techniques. Alon and Tarsi used polynomials to prove several conjectures about the chromatic number of a graph [2] They provided equivalent conditions for a graph to be not k-colorable using polynomial ideals. De Loera et al and Hillar proved results concerning the algebraic characterization of a graph colorability [7], [12]. The main tools de Loera et al use in their algebraic characterizations for the colorability of a graph are polynomial ideals and Grobner bases. The main result of this paper, Theorem 2, extends the following result by Hillar and Windfeldt [12] to uniform hypergraphs. Our proofs can be extended to determine the existence of 2-colorings of hypergraphs satisfying specified color patterns introduced in the subsection
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