Depths and Cohen–Macaulay properties of path ideals

Given a tree T on n vertices, there is an associated ideal I   of R[x1,…,xn]R[x1,…,xn] generated by all paths of a fixed length ℓ of T  . We classify all trees for which R/IR/I is Cohen–Macaulay, and we show that an ideal I whose generators correspond to any collection of subtrees of T satisfies the König 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.