Let be a weighted oriented graph and let I() be its edge ideal. Under a natural condition that the underlying (undirected) graph of contains a perfect matching consisting of leaves, we provide several equivalent conditions for the Cohen-Macaulayness of I(). We also completely characterize the Cohen-Macaulayness of I() when the underlying graph of is a bipartite graph. When I() fails to be Cohen-Macaulay, we give an instance where I() is shown to be sequentially Cohen-Macaulay.
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