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

TL;DR

This paper explores the relationship between hypergraph colorings, perfect graphs, and algebraic properties of monomial ideals, providing new algebraic characterizations of perfect graphs and methods to analyze associated primes of ideal powers.

Contribution

It offers explicit descriptions of associated primes of powers of cover ideals and introduces algebraic methods to determine hypergraph chromatic numbers, leading to new characterizations of perfect graphs.

Findings

Explicit description of associated primes of J^s in terms of hypergraph colorings

Algebraic method for computing hypergraph chromatic number

Two new algebraic characterizations of perfect graphs

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.