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Sequentially Cohen-Macaulay edge ideals

2007; American Mathematical Society; Volume: 135; Issue: 08 Linguagem: Inglês

10.1090/s0002-9939-07-08841-7

1088-6826

Christopher A. Francisco, Adam Van Tuyl,

Polynomial and algebraic computation

Let $G$ be a simple undirected graph on $n$ vertices, and let $\mathcal I(G) \subseteq R = k[x_1,\ldots ,x_n]$ 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 $\mathcal I(G)$ 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.