Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers
By Huy Tài Hà, Adam Van Tuyl. 2006
277 citations recorded by the source corpus.
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 sub-hypergraphs 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.
Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/c3bfd2f124c4a0871f4b7a4be942c45cd20d26cd
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In Algebra and Number Theory
In Computational Theory and Mathematics
In Geometry and Topology
Citations in the corpus
- Non-vanishingness of Betti numbers of edge ideals (2011, methodology): “In Section 2, we recall some definitions on graphs and results by Zheng [16], Hà and Van Tuyl [7], and Katzman [11] mentioned above.”
- A CLASS OF HYPERGRAPHS THAT GENERALIZES CHORDAL GRAPHS (2008, background): “In Theorem-definition 2.1 we show that the properties of being triangulated, chordal, and having a perfect elimination order, are equivalent also for hypergraphs.”
- Matchings, coverings, and Castelnuovo-Mumford regularity (2010, background): “We recover and extend results of [18] and [24], but show that results of [26] and [33] cannot be shown using this technique.”
- Splittings of monomial ideals (2008, unknown): “A C ] 1 3 Fe b 20 09”
- Regularity of powers of bipartite graphs (2016, unknown): “In particular, there has been active research on bounding the homological invariants of edge ideals in terms of the combinatorial invariants of the associated graphs, see for example [3,10,12,15,17,19,22,23,26,28,32,33].”