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• W2950456445 abstract "Given a tree T on n vertices, there is an associated ideal I of a polynomial ring in n variables over a field, generated by all paths of a fixed length of T. We show that such an ideal always satisfies the Konig property and classify all trees for which R/I is Cohen-Macaulay. More generally, we show that an ideal I whose generators correspond to any collection of subtrees of T satisfies the Konig property. Since the edge ideal of a simplicial tree has this form, this generalizes a result of Faridi. Moreover, every square-free monomial ideal can be represented (non-uniquely) as a subtree ideal of a graph, so this construction provides a new combinatorial tool for studying square-free monomial ideals. For a special class of trees, namely trees that are themselves a path, a precise formula for the depth is given and it is shown that the proof extends to provide a lower bound on the Stanley depth of these ideals. Combining these results gives a new class of ideals for which the Stanley Conjecture holds." @default.
• W2950456445 created "2019-06-27" @default.
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• W2950456445 date "2012-11-20" @default.
• W2950456445 modified "2023-09-25" @default.
• W2950456445 title "Depths and Cohen-Macaulay Properties of Path Ideals" @default.
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• W2950456445 hasPublicationYear "2012" @default.
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