Related papers: Balanced generalized lower bound inequality for si…

We introduce the property of convex normality of rational polytopes and give a dimensionally uniform lower bound for the edge lattice lengths, guaranteeing the property. As an application, we show that if every edge of a lattice d-polytope…

Denham, Suciu and Panov, Ray computed ranks of homotopy groups and Poincar\'e series of a moment-angle-complex $\mathcal Z(\mathcal K$) / Davis-Januzskiewicz space $DJ(\mathcal K)$ associated to a flag simplicial complex $\mathcal K$. In…

Kalai proved that the simplicial polytopes with g_2=0 are the stacked polytopes. We characterize the g_2=1 case. Specifically, we prove that every simplicial d-polytope (d>=4) which is prime and with g_2=1 is combinatorially equivalent…

Let P be a random $d$-dimensional 0/1-polytope with $n(d)$ vertices, and denote by $\phi_k(P)$ the \emph{$k$-face density} of $P$, i.e., the quotient of the number of $k$-dimensional faces of $P$ and $\binom{n(d)}{k+1}$. For each $k\ge 2$,…

We prove that for any convex polytope $\Omega \subset \mathbb{R}^d$ which is centrally symmetric and whose faces of all dimensions are also centrally symmetric, there exists a Riesz basis of exponential functions in the space $L^2(\Omega)$.…

Neighborly cubical polytopes are known as the cubical analogues of the cyclic polytopes. Using the short cubical $h$-vectors of cubical polytopes (introduced by Adin), we derive an explicit formula for the face numbers of the neighborly…

We prove tight lower bounds for the coefficients of the generalized $h$-vector of a rational polytope with a symmetry of prime order that is fixed--point--free on the boundary. These bounds generalize results of R.~Stanley and R.~Adin for…

We resolve a conjecture of Kalai asserting that the $g_2$-number of any simplicial complex $\Delta$ that represents a connected normal pseudomanifold of dimension $d\geq 3$ is at least as large as ${d+2 \choose 2}m(\Delta)$, where…

In [Ho] A.J. Hoffman proved a lower bound on the chromatic number of a graph in the terms of the largest and the smallest eigenvalues of its adjacency matrix. In this paper, we prove a higher dimensional version of this result and give a…

In this paper, we explore algebraic approaches to $d$-improper and $t$-clustered colourings, where the colouring constraints are relaxed to allow some monochromatic edges. Bilu [J. Comb. Theory Ser. B, 96(4):608-613, 2006] proved a…

Chromatic symmetric functions are well-studied symmetric functions in algebraic combinatorics that generalize the chromatic polynomial and are related to Hessenberg varieties and diagonal harmonics. Motivated by the Stanley--Stembridge…

In 1967, Gr\"unbaum conjectured that the function $$ \phi_k(d+s,d):=\binom{d+1}{k+1}+\binom{d}{k+1}-\binom{d+1-s}{k+1},\; \text{for $2\le s\le d$} $$ provides the minimum number of $k$-faces for a $d$-dimensional polytope (abbreviated as a…

According to Euler's relation any polytope P has as many faces of even dimension as it has faces of odd dimension. As a generalization of this fact one can compare the number of faces whose dimension is congruent to i modulo m with the…

Given a shifted order ideal $U$, we associate to it a family of simplicial complexes $(\Delta_t(U))_{t\geq 0}$ that we call squeezed complexes. In a special case, our construction gives squeezed balls that were defined and used by Kalai to…

The Grishukhin inequality Gr_7 is a facet of CutP_7, the cut polytope on seven points, which is ``sporadic'' in the sense that its proper generalization has not been known. In this paper, we extend Gr_7 to an inequality I(G,H) valid for…

The family of Buchsbaum simplicial posets generalizes the family of simplicial cell manifolds. The $h'$-vector of a simplicial complex or simplicial poset encodes the combinatorial and topological data of its face numbers and the reduced…

We prove that if a pure simplicial complex of dimension d with n facets has the least possible number of (d-1)-dimensional faces among all complexes with n faces of dimension d, then it is vertex decomposable. This answers a question of J.…

A conjecture of Kalai asserts that for $d\geq 4$, the affine type of a prime simplicial $d$-polytope $P$ can be reconstructed from the space of affine $2$-stresses of $P$. We prove this conjecture for all $d\geq 5$. We also prove the…

Generalized permutahedra are the polytopes obtained from the permutahedron by changing the edge lengths while preserving the edge directions, possibly identifying vertices along the way. We introduce a "lifting" construction for these…

Let P be a d-dimensional lattice polytope. We show that there exists a natural number c_d, only depending on d, such that the multiples cP have a unimodular cover for every natural number c >= c_d. Actually, a subexponential upper bound for…