Cohen-Macaulay Weighted Oriented Edge Ideals and its Alexander Dual

TL;DR

This paper extends Cohen-Macaulay properties to weighted oriented edge ideals of graphs, classifies such ideals for cycles, and explores their Alexander duals, confirming a conjecture for cycle graphs.

Contribution

It generalizes Cohen-Macaulay constructions from simple graphs to weighted oriented graphs and classifies these ideals for cycle graphs, also analyzing their Alexander duals.

Findings

Cohen-Macaulay classification for cycle graphs

Validation of the conjecture on Cohen-Macaulayness for cycles

Conditions for Alexander duals to be Cohen-Macaulay

Abstract

The study of the edge ideal $I(D_{G})$ of a weighted oriented graph $D_{G}$ with underlying graph $G$ started in the context of Reed-Muller type codes. We generalize a Cohen-Macaulay construction for $I(D_{G})$, which Villarreal gave for edge ideals of simple graphs. We use this construction to classify all the Cohen-Macaulay weighted oriented edge ideals, whose underlying graph is a cycle. We show that the conjecture on Cohen-Macaulayness of $I(D_{G})$, proposed by Pitones et al. (2019), holds for $I(D_{C_{n}})$, where $C_{n}$ denotes the cycle of length $n$. Miller generalized the concept of Alexander dual ideals of square-free monomial ideals to arbitrary monomial ideals, and in that direction, we study the Alexander dual of $I(D_{G})$ and its conditions to be Cohen-Macaulay.