The theory of complexes of directed trees was initiated by Kozlov to answer a question by Stanley, and later on, results from the theory were used by Babson and Kozlov in their proof of the Lovász conjecture. We develop the theory and prove that complexes on directed acyclic graphs are shellable. A related concept is that of independence complexes: construct a simplicial complex on the vertex s...

In this paper, we introduce a subclass of chordal graphs which contains $d$-trees and show that their complement are vertex decomposable and so is shellable and sequentially Cohen-Macaulay.

The augmented Bergman complex of a matroid is simplicial introduced recently in work Braden, Huh, Matherne, Proudfoot and Wang. It may be viewed as hybrid two well-studied pure shellable complexes associated to matroids: the independent set complex.
 shown here that also shellable, via different families shelling orders. Furthermore, comparing description its homotopy type induced from she...

Preface Everything started from one book. I happened to buy the textbook \Lectures on Poly-topes" 98] written by Prof. G unter M. Ziegler, at the university bookstore about ve years ago. I bought it only because the gures (especially of permutahedra and of zonotopal tilings) interested me, but the book turned out to be a very good introduction to the world of poly-topes, starting from fundament...

The flag f-vectors of three-colored complexes are characterized. This also characterizes the flag h-vectors of balanced Cohen-Macaulay complexes of dimension two, as well as the flag h-vectors of balanced shellable complexes of dimension two.

The concept of shellability of complexes is generalized by deleting the requirement of purity (i.e., that all maximal faces have the same dimension). The usefulness of this level of generality was suggested by certain examples coming from the theory of subspace arrangements. We develop several of the basic properties of the concept of nonpure shellability. Doubly indexed f -vectors and h-vector...

The set of all permutations, ordered by pattern containment, forms a poset. This paper presents the first explicit major results on the topology of intervals in this poset. We show that almost all (open) intervals in this poset have a disconnected subinterval and are thus not shellable. Nevertheless, there seem to be large classes of intervals that are shellable and thus have the homotopy type ...

All dissections of a convex (mn + 2)-gons into (m + 2)-gons are facets of a simplicial complex. This complex is introduced by S. Fomin and A.V. Zelevinsky in [7]. We reprove the result of E. Tzanaki about shellability of such complex by finding a concrete shelling order. Also, we use this shelling order to find a combinatorial interpretation of h-vector and to describe the generating facets of ...