Edge ideals of oriented graphs

TL;DR

This paper characterizes when the edge ideals of weighted oriented graphs are Cohen-Macaulay, providing equivalent conditions and complete characterizations for bipartite underlying graphs, and discusses sequential Cohen-Macaulayness in non-Cohen-Macaulay cases.

Contribution

It offers new criteria and complete characterizations for Cohen-Macaulayness of edge ideals of weighted oriented graphs, especially for bipartite underlying graphs.

Findings

Equivalent conditions for Cohen-Macaulayness under a natural leaf-matching condition.

Complete characterization of Cohen-Macaulay edge ideals when the underlying graph is bipartite.

Examples of sequential Cohen-Macaulayness when Cohen-Macaulayness fails.

Abstract

Let $\mathcal{D}$ be a weighted oriented graph and let $I(\mathcal{D})$ be its edge ideal. Under a natural condition that the underlying (undirected) graph of $\mathcal{D}$ contains a perfect matching consisting of leaves, we provide several equivalent conditions for the Cohen-Macaulayness of $I(\mathcal{D})$. We also completely characterize the Cohen-Macaulayness of $I(\mathcal{D})$ when the underlying graph of $\mathcal{D}$ is a bipartite graph. When $I(\mathcal{D})$ fails to be Cohen-Macaulay, we give an instance where $I(\mathcal{D})$ is shown to be sequentially Cohen-Macaulay.