Sequentially Cohen-Macaulay edge ideals

Let GG be a simple undirected graph on nn vertices, and let I(G)⊆R=k[x1,…,xn]\mathcal I(G) \subseteq R = k[x_1,\ldots ,x_n] denote its associated edge ideal. We show that all chordal graphs GG are sequentially Cohen-Macaulay; our proof depends upon showing that the Alexander dual of I(G)\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.