We use the correspondence between hypergraphs and their associated edge ideals to study the minimal graded free resolution of squarefree monomial ideals. The theme of this paper is to understand how the combinatorial structure of a hypergraph H appears within the resolution of its edge ideal I(H). We discuss when recursive formulas to compute the graded Betti numbers of I(H) in terms of its subhypergraphs can be obtained; these results generalize our previous work (Hà, H.T., Van Tuyl, A. in J. Algebra 309:405-425, 2007) on the edge ideals of simple graphs. We introduce a class of hypergraphs, which we call properly-connected, that naturally generalizes simple graphs from the point of view that distances between intersecting edges are "well behaved." For such a hypergraph H (and thus, for any simple graph), we give a lower bound for the regularity of I(H) via combinatorial information describing H and an upper bound for the regularity when H = G is a simple graph. We also introduce triangulated hypergraphs that are properly-connected hypergraphs generalizing chordal graphs. When H is a triangulated hypergraph, we explicitly compute the regularity of I(H) and show that the graded Betti numbers of I(H) are independent of the ground field. As a consequence, many known results about the graded Betti numbers of forests can now be extended to chordal graphs.

Abstract. We provide some new conditions under which the graded Betti numbers of a monomial ideal can be computed in terms of the graded Betti numbers of smaller ideals, thus complementing Eliahou and Kervaire's splitting approach. As applications, we show that edge ideals of graphs are splittable, and we provide an iterative method for computing the Betti numbers of the cover ideals of Cohen-Macaulay bipartite graphs. Finally, we consider the frequency with which one can find particular splittings of monomial ideals and raise questions about ideals whose resolutions are characteristic-dependent.

Associated to a simple undirected graph G is a simplicial complex Δ G whose faces correspond to the independent sets of G. We call a graph G shellable if Δ G is a shellable simplicial complex in the nonpure sense of Björner-Wachs. We are then interested in determining what families of graphs have the property that G is shellable. We show that all chordal graphs are shellable. Furthermore, we classify all the shellable bipartite graphs; they are precisely the sequentially Cohen-Macaulay bipartite graphs. We also give a recursive procedure to verify if a bipartite graph is shellable. Because shellable implies that the associated Stanley-Reisner ring is sequentially Cohen-Macaulay, our results complement and extend recent work on the problem of determining when the edge ideal of a graph is (sequentially) Cohen-Macaulay. We also give a new proof for a result of Faridi on the sequentially Cohen-Macaulayness of simplicial forests.

Abstract. Let G be a simple undirected graph on n vertices, and let I(G) ⊆ R = k[x 1 , . . . , x n ] denote its associated edge ideal. We show that all chordal graphs G are sequentially Cohen-Macaulay; our proof depends upon showing that the Alexander dual of I(G) is componentwise linear. Our result complements Faridi's theorem that the facet ideal of a simplicial tree is sequentially Cohen-Macaulay and implies Herzog, Hibi, and Zheng's theorem that a chordal graph is Cohen-Macaulay if and only if its edge ideal is unmixed. We also characterize the sequentially Cohen-Macaulay cycles and produce some examples of nonchordal sequentially Cohen-Macaulay graphs.

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.