Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers

Abstract

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.

A class of hypergraphs that generalizes chordal graphs

“…See for instance [5], [7], [10], [11], [12], [13], [14], [16], [19]. In [10], the authors use certain connectedness properties to determine a class of hypergraphs such that the hypergraph algebras have linear resolutions. Furthermore, nice recursive formulas for computing the Betti numbers are given.…”

“…This definition is basically due to Hà and Van Tuyl, see [10] Definition 5.5. However, in [10] the property being triangulated is defined only on a special class of hypergraphs called properly-connected.…”

“…However, in [10] the property being triangulated is defined only on a special class of hypergraphs called properly-connected. For a further discussion see Section 2.2 below.…”

“…As mentioned after Definition 2.5, the concept of triangulated hypergraph also occurs in [10]. There the authors (among other things) aim for a generalization of Fröberg's theorem (Theorem 1.1).…”

Regularity of path ideals of gap free graphs

“…People have studied the regularity of the edge ideals of hypergraphs and their relation with the combinatorics of the hypergraph. We refer to [9] and [10] for studies in that area.…”