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arXiv · 0908.1505

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

Christopher A. Francisco·Huy Tai Ha·Adam Van Tuyl
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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.

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Christopher A. Francisco, Huy Tai Ha, Adam Van Tuyl. 2009-08-11. Colorings of hypergraphs, perfect graphs, and associated primes of powers of monomial ideals. https://arxiv.org/abs/0908.1505

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