The maximum diameter of pure simplicial complexes and pseudo-manifolds

Vertex Decomposable Simplicial Complexes Associated to Path Graphs

Introduction Vertex decomposability of a simplicial complex is a combinatorial topological concept which is related to the algebraic properties of the Stanley-Reisner ring of the simplicial complex. This notion was first defined by Provan and Billera in 1980 for k-decomposable pure complexes which is known as vertex decomposable when . Later Bjorner and Wachs extended this concept to non-pure ...

Recent Progress on the Diameter of Polyhedra and Simplicial Complexes

Some recent attempts of settling this question go by looking at the problem in the more general context of pure simplicial complexes: What is the maximum diameter of the dual graph of a simplicial (d − 1)sphere or (d− 1)-ball with n vertices? Here a simplicial (d − 1)-ball or sphere is a simplicial complex homeomorphic to the (d − 1)-ball or sphere. These complexes are necessarily pure (all the...

On the Multiplicity Conjecture for Non-cohen-macaulay Simplicial Complexes

We prove a reformulation of the multiplicity upper bound conjecture and use that reformulation to prove it for three-dimensional simplicial complexes and homology manifolds with many vertices. We provide necessary conditions for a Cohen-Macaulay complex with many vertices to have a pure minimal free resolution and a characterization of flag complexes whose minimal free resolution is pure.

On Isomorphism of Simplicial Complexes and Their Related Algebras

Projectivities in Simplicial Complexes and Colorings of Simple Polytopes

For each strongly connected finite-dimensional (pure) simplicial complex ∆ we construct a finite group Π(∆), the group of projectivities of ∆, which is a combinatorial but not a topological invariant of ∆. This group is studied for combinatorial manifolds and, in particular, for polytopal simplicial spheres. The results are applied to a coloring problem for simplicial (or, dually, simple) polyt...