Colorings of hypergraphs, perfect graphs, and associated primes of powers of monomial ideals

Abstract

There is a natural one-to-one correspondence between squarefree monomial ideals and finite simple hypergraphs via the cover ideal construction. Let H be a finite simple hypergraph, and let J = J ( H ) be its cover ideal in a polynomial ring R. We give an explicit description of all associated primes of R / J s , for any power J s of J, in terms of the coloring properties of hypergraphs arising from H . We also 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. 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 / J s ) , while the second characterization is in terms of the saturated chain condition for associated primes.