Cohen-Macaulayness of Powers of Edge Ideals of Weighted Oriented Graphs

For the edge ideal $I(\text{D})$ of a weighted oriented graph $\text{D}$, we prove that its symbolic powers $I(\text{D}{)}^{(t)}$ are Cohen-Macaulay for all $t⩾1$ if and only if the underlying graph $G$ is composed of a disjoint union of some complete graphs. We also completely characterize the Cohen-Macaulayness of the ordinary powers $I(\text{D}{)}^t$ for all $t⩾2$. Furthermore, we provide a criterion for determining whether $I(\text{D}{)}^t=I(\text{D}{)}^{(t)}$.