  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 sub-hypergraphs can be obtained; these results generalize
our previous work 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, a properly-connected
hypergraph which is a generalization of 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.
