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/Js, for any power Js 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/Js), while the second characterization is in terms of the saturated chain condition for associated primes. © 2010 Elsevier Inc.