Depths and Cohen-Macaulay Properties of Path Ideals

TL;DR

This paper investigates the algebraic properties of path ideals of trees, establishing conditions for Cohen-Macaulayness, the Konig property, and providing formulas for depth and Stanley depth, thus contributing new combinatorial tools for monomial ideal analysis.

Contribution

It classifies trees with Cohen-Macaulay path ideals, generalizes the Konig property to subtree ideals, and links these to Stanley depth and the Stanley Conjecture.

Findings

Path ideals of trees satisfy the Konig property.

Classification of trees with Cohen-Macaulay path ideals.

A formula for depth of path ideals of path trees.

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.