  Herzog, Hibi, and Zheng classified the Cohen-Macaulay edge ideals of chordal
graphs. In this paper, we classify Cohen-Macaulay edge ideals of (vertex)
weighted oriented chordal and simplicial graphs, a more general class of
monomial ideals. In particular, we show that the Cohen-Macaulay property of
these ideals is equivalent to the unmixed one and hence, independent of the
underlying field.
