Sequentially Cohen-Macaulay edge ideals

By Christopher A. Francisco, Adam Van Tuyl. 2005

146 citations recorded by the source corpus.

Let G be a simple undirected graph on n vertices, and let I(G) C R = k[x 1 ,...,x n ] denote its associated edge ideal. We show that all chordal graphs G are sequentially Cohen-Macaulay; our proof depends upon showing that the Alexander dual of I(G) is componentwise linear. Our result complements Faridi's theorem that the facet ideal of a simplicial tree is sequentially Cohen-Macaulay and implies Herzog, Hibi, and Zheng's theorem that a chordal graph is Cohen-Macaulay if and only if its edge ideal is unmixed. We also characterize the sequentially Cohen-Macaulay cycles and produce some examples of nonchordal sequentially Cohen-Macaulay graphs.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Pharmacology

View Paper: https://www.semanticscholar.org/paper/110a5080d87704040e88c191a771bf1fd33c4026

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Citations in the corpus

Splittable ideals and the resolutions of monomial ideals (2005, background): “The references [3, 5, 9, 10, 11, 13, 14, 17, 18, 19, 21, 23, 24, 25, 26, 27] form a partial list of references on this topic.”
Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers (2006, background): “Subsequently, many people, including [1, 12, 13, 15, 16, 17, 18, 27, 29, 28, 30, 32], have been working on a program to build a dictionary between the algebraic properties of I(G) and the combinatorial structure of G.”
Regularity of Edge Ideals and Their Powers (2017, background): “e [2]). x y z w t Figure 12. A vertex decomposable and a (weakly) chordal graph Remark 4.13. Chordal graphs are vertex decomposable and vertex decomposable graphs are sequentially Cohen-Macaulay (see [26, 65]). Sequentially Cohen-Macaulay bipartite graphs are vertex decomposable (see [62]). Also if ”
Edge ideals: algebraic and combinatorial properties (2010, background): “[43] The only sequentially Cohen-Macaulay cycles are C3 and C5”
Sequentially Cohen–Macaulay bipartite graphs: vertex decomposability and regularity (2009, background): “Trees, which are always bipartite, were first shown to be SCM in [6]; alternative proofs were given in [4, 9, 20, 24].”
Resolutions of square-free monomial ideals via facet ideals: a survey (2006, background): “To learn more about the properties of edge ideals, one should see [1, 2, 6, 16, 17, 25, 26, 27, 36, 38, 39, 40, 41, 44, 46]; further properties of facet ideals can be found in the sequels [12, 13, 14] to Faridi’s paper cited above.”
Bounds on the regularity and projective dimension of ideals associated to graphs (2011, background): “It should be noted that the projective dimension and regularity of Stanley-Reisner ideals, or more specifically edge ideals, have attracted a lot of attention; see [3, 10, 11, 12, 16, 21, 23] and references given therein.”
Splittings of monomial ideals (2008, background): “Identifying classes of Cohen-Macaulay graphs is a topic of much interest [6, 7, 11, 16].”