Mathematics – Combinatorics
Scientific paper
2011-09-21
Mathematics
Combinatorics
This is a new version in which Section 3 about complexes $\mathcal{C}_n^k$ is removed. There are some troubles in the proof of
Scientific paper
The question of shellability of complexes of directed trees was asked by R. Stanley. D. Kozlov showed that the existence of a complete source in a directed graph provides a shelling of its complex of directed trees. We will show that this property gives a shelling that is straightforward in some sense. Among the simplicial polytopes, only the crosspolytopes allow such a shelling. Furthermore, we show that the complex of directed trees of a complete double directed graph is a union of suitable spheres. We also investigate shellability of the maximal pure skeleton of a complex of directed trees. Also, we prove that is vertex-decomposable. For these complexes we describe the set of generating facets.
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