Cohen-Macaulay Weighted Oriented Chordal and Simplicial Graphs

TL;DR

This paper extends the classification of Cohen-Macaulay edge ideals to weighted oriented chordal and simplicial graphs, showing their Cohen-Macaulay property is equivalent to being unmixed and field-independent.

Contribution

It provides a classification of Cohen-Macaulay edge ideals for a broader class of weighted oriented graphs, generalizing previous results.

Findings

Cohen-Macaulay property is equivalent to unmixedness for these graphs.

The Cohen-Macaulay property is independent of the underlying field.

The classification applies to weighted oriented chordal and simplicial graphs.

Abstract

Herzog, Hibi, and Zheng classified the Cohen-Macaulay edge ideals of chordal graphs. In this paper, we classify Cohen-Macaulay edge ideals of (vertex) weighted oriented chordal and simplicial graphs, a more general class of monomial ideals. In particular, we show that the Cohen-Macaulay property of these ideals is equivalent to the unmixed one and hence, independent of the underlying field.