Colorings of Hypergraphs, Perfect Graphs, and Associated Primes of Powers of Monomial Ideals

Let H denote a finite simple hypergraph. The cover ideal of H, denoted by J = J(H), is the monomial ideal whose minimal generators correspond to the minimal vertex covers of H. We give an algebraic method for determining the chromatic number of H, proving that it is equivalent to a monomial ideal membership problem involving powers of J . Furthermore, we study the sets Ass(R/Js) by exploring the rôle played by the coloring properties of induced subhypergraphs of H. Our work yields two new purely algebraic characterizations of perfect graphs, independent of the Strong Perfect Graph Theorem; the first characterization is in terms of the sets Ass(R/Js), while the second characterization is in terms of the saturated chain condition for associated primes.