Complexes of Directed Trees

To every directed graph G one can associate a complex (G) consisting of directed subforests. This construction, suggested to us by R. Stanley, is especially important in the case of a complete double directed graph Gn, where it leads to studying some interesting representations of the symmetric group and corresponds (via Stanley-Reisner correspondence) to an interesting quotient ring. Our main result states that (Gn) is shellable, in particular Cohen-Macaulay, which can be further translated to say that the Stanley-Reisner ring of (Gn) is Cohen-Macaulay. Besides that, by computing the homolo-gy groups of (G) for the cases when G is essentially a tree and when G is a double directed cycle, we touch upon the general question of the interaction of the combinatorial properties of a graph and topological properties of the associated complex.