Dirac’s theorem on chordal graphs and Alexander duality

Herzog, Jürgen; Hibi, Takayuki; Zheng, Xinxian

doi:10.1016/j.ejc.2003.12.008

Cited by 82 publications

(70 citation statements)

References 8 publications

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“…Triangulated hypergraphs generalize the notion of chordal graphs, which has attracted considerable attention lately (cf. [12,13,17,18]). In fact, triangulated graphs are precisely chordal graphs.…”

Section: Theorem 11 (Theorem 32) Let H Be a Hypergraph With Two Or mentioning

confidence: 99%

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

2007

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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.

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