Triangle-free Cohen–Macaulay oriented graph conjecture

Let $\operatorname{\mathcal{D}}$ be a weighted oriented graph whose underlying graph $G$ is triangle-free, meaning that $G$ contains no cycles of length $3$. The graph $G$ is Cohen–Macaulay when its edge ideal is Cohen–Macaulay, and $\operatorname{\mathcal{D}}$ is unmixed when all of its minimal vertex covers have the same cardinality. Triangle-free Cohen–Macaulay oriented graph conjecture. $\operatorname{\mathcal{D}}$ is Cohen–Macaulay if and only if $G$ is Cohen–Macaulay and $\operatorname{\mathcal{D}}$ is unmixed. This is proposed based on the preceding theorem and the authors' computations; its status remains open in the source.