Sequentially Cohen-Macaulay edge ideals

Abstract:

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.