Recent Progress on the Diameter of Polyhedra and Simplicial Complexes
نویسنده
چکیده
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 maximal simplices have the same dimension). The dual graph of a pure simplicial complex is the graph whose vertices are the maximal simplices (a. k. a. facets) and whose edges correspond to adjacent facets. We can also remove the sphere/ball condition and ask the same for all pure simplicial complexes. Some recent results in this direction are:
منابع مشابه
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 ...
متن کاملCohen-Macaulay-ness in codimension for simplicial complexes and expansion functor
In this paper we show that expansion of a Buchsbaum simplicial complex is $CM_t$, for an optimal integer $tgeq 1$. Also, by imposing extra assumptions on a $CM_t$ simplicial complex, we provethat it can be obtained from a Buchsbaum complex.
متن کاملNew methods for constructing shellable simplicial complexes
A clutter $mathcal{C}$ with vertex set $[n]$ is an antichain of subsets of $[n]$, called circuits, covering all vertices. The clutter is $d$-uniform if all of its circuits have the same cardinality $d$. If $mathbb{K}$ is a field, then there is a one-to-one correspondence between clutters on $V$ and square-free monomial ideals in $mathbb{K}[x_1,ldots,x_n]$ as follows: To each clutter $mathcal{C}...
متن کاملOn a special class of Stanley-Reisner ideals
For an $n$-gon with vertices at points $1,2,cdots,n$, the Betti numbers of its suspension, the simplicial complex that involves two more vertices $n+1$ and $n+2$, is known. In this paper, with a constructive and simple proof, wegeneralize this result to find the minimal free resolution and Betti numbers of the $S$-module $S/I$ where $S=K[x_{1},cdots, x_{n}]$ and $I$ is the associated ideal to ...
متن کامل