A recent conjecture that appeared in three papers by Bigdeli–Faridi, Dochtermann, and Nikseresht, is every simplicial complex whose clique has shellable Alexander dual, ridge-chordal. This strengthens the long-standing Simon's k-skeleton of simplex extendably shellable, for any k. We show stronger a negative answer, exhibiting an infinite family counterexamples.

We prove that the (d − 2)-nd barycentric subdivision of every convex d-ball is shellable. This yields a new characterization of the PL property in terms of shellability: A sphere or a ball is PL if and only if it becomes shellable after sufficiently many barycentric subdivisions. This improves results by Whitehead, Zeeman and Glaser. Moreover, we show the Zeeman conjecture is equivalent to the ...

We show that if a three-dimensional polytopal complex has a knot in its 1-skeleton, where the bridge index of the knot is larger than the number of edges of the knot, then the complex is not constructible, and hence, not shellable. As an application we settle a conjecture of Hetyei concerning the shellability of cubical barycentric subdivisions of 3-spheres. We also obtain similar bounds conclu...

Let Φ be a finite root system of rank n and let m be a positive integer. It is proved that the generalized cluster complex ∆m(Φ), introduced by S. Fomin and N. Reading, is (m + 1)-Cohen-Macaulay, in the sense of Baclawski. This statement was conjectured by V. Reiner. More precisely, it is proved that the simplicial complex obtained from ∆m(Φ) by removing any subset of its vertex set of cardinal...

We give new examples of shellable but not extendably shellable two dimensional simplicial complexes. They include minimal examples, which are smaller than those previously known. We also give examples of shellable but not vertex decomposable two dimensional simplicial complexes. Among them are extendably shellable ones. This shows that neither extendable shellability nor vertex decomposability ...

We introduce a new class of lattices, the modernistic lattices, and their duals, the comodernistic lattices. We show that every modernistic or comodernistic lattice has shellable order complex. We go on to exhibit a large number of examples of (co)modernistic lattices. We show comodernism for two main families of lattices that were not previously known to be shellable: the order congruence latt...

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.

We investigate the shellability of polyhedral join $\mathcal{Z}^*_M (K, L)$ simplicial complexes $K, M$ and a subcomplex $L \subset K$. give sufficient conditions necessary on $(K, for being shellable. In particular, we show that some pairs L)$, becomes shellable regardless whether $M$ is or not. Polyhedral joins can be applied to graph theory as independence complex certain generalized version...

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...