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

For the edge ideal $I(\D)$ of a weighted oriented graph $\D$, we prove that its symbolic powers $I(\D)^{(t)}$ are Cohen-Macaulay for all $t\geqslant 1$ 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 $I(\D)^t$ for all $t\geqslant 2$. Furthermore, we provide a criterion for determining whether $I(\D)^t=I(\D)^{(t)}$.