Unmixed and Cohen--Macaulay weighted oriented K\"onig graphs

TL;DR

This paper characterizes when the edge ideal of a weighted oriented graph is unmixed or Cohen--Macaulay based on the absence of certain cycles and properties of the underlying graph, such as being K"onig.

Contribution

It provides new characterizations of unmixed and Cohen--Macaulay properties for edge ideals of weighted oriented graphs under specific cycle restrictions.

Findings

Unmixedness characterized for graphs with no 3, 5, or 7 cycles or K"onig graphs.

Cohen--Macaulayness characterized for graphs with no 3 or 5 cycles or K"onig graphs.

Unmixedness and Cohen--Macaulayness coincide for graphs with girth greater than 7 or K"onig graphs without 4-cycles.

Abstract

Let $D$ be a weighted oriented graph, whose underlying graph is $G$, and let $I(D)$ be its edge ideal. If $G$ has no $3$-, $5$-, or $7$-cycles, or $G$ is K\"{o}nig, we characterize when $I(D)$ is unmixed. If $G$ has no $3$- or $5$-cycles, or $G$ is K\"onig, we characterize when $I(D)$ is Cohen--Macaulay. We prove that $I(D)$ is unmixed if and only if $I(D)$ is Cohen--Macaulay when $G$ has girth greater than $7$ or $G$ is K\"onig and has no $4$-cycles.