Cohen–Macaulay chordal graphs

Herzog, Jürgen; Hibi, Takayuki; Zheng, Xinxian

doi:10.1016/j.jcta.2005.08.007

Cited by 66 publications

(61 citation statements)

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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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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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“…Chordal graphs have been in particular focus and indeed, many natural results and questions seem to be connected to this class of graphs. See for example [3], [14], [15], [17], [18], [11]. More recently, also edge ideals of hypergraphs have been studied and many results and familiar properties of graphs have got hypergraph analogue.…”

Section: Introductionmentioning

confidence: 99%

Some algebraic properties of hypergraphs

2011

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“…Gorenstein) graphs. This problem was considered for certain classes of graphs (see [5,6,9,10]). Generally, we cannot read off the Cohen-Macaulay and Gorenstein properties of G just from its structure because these properties in fact depend on the characteristic of the base field K (see [20, Exercise 5.3.31] and [10, Proposition 2.1]).…”

Section: Introductionmentioning

confidence: 99%

Buchsbaumness of the second powers of edge ideals

Hoang

Trung

2018

J. Algebra Appl.

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Abstract. We graph-theoretically characterize the class of graphs G such that I(G) 2 are Buchsbaum. IntroductionThroughout this paper let G = (V (G), E(G)) be a finite simple graph without isolated vertices. An independent set in G is a set of vertices no two of which are adjacent to each other. The size of the largest independent set, denoted by α(G), is called the independence number of G. A graph is called well-covered if every maximal independent set has the same size. A well-covered graph G is a member of the class W 2 if the remove any vertex of G leaves a well-covered graph with the same independence number as G (see e.g. [14]).Let R = K[x 1 , . . . , x n ] be a polynomial ring of n variables over a given field K. Let G be a simple graph on the vertex set V (G) = {x 1 , . . . , x n }. We associate to the graph G a quadratic squarefree monomial idealwhich is called the edge ideal of G. We say that G is Cohen-Macaulay (resp. Gorenstein) if I(G) is a Cohen-Macaulay (resp. Gorenstein) ideal. It is known that G is well-covered whenever it is Cohen-Macaulay (see e.g. [20, Proposition 6.1.21]) and G is in W 2 whenever it is Gorenstein (see e.g. [10, Lemma 2.5]). It is a wide open problem to characterize graph-theoretically the Cohen-Macaulay (resp. Gorenstein) graphs. This problem was considered for certain classes of graphs (see [5,6,9,10]). Generally, we cannot read off the Cohen-Macaulay and Gorenstein properties of G just from its structure because these properties in fact depend on the characteristic of the base field K (see [20, Exercise 5.3.31] and [10, Proposition 2.1]).If we move on to the higher powers of I(G), then we can graph-theoretically characterize G such that I(G) m is Cohen-Macaulay (or Buchsbaum, or generalized CohenMacaulay) for some m 3 (and for all m 1) (see [4,15,19]

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Cohen–Macaulay chordal graphs

Abstract: We classify all Cohen-Macaulay chordal graphs. In particular, it is shown that a chordal graph is Cohen-Macaulay if and only if it is unmixed.

Cited by 66 publications

References 4 publications

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

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

Some algebraic properties of hypergraphs

Buchsbaumness of the second powers of edge ideals

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