Related papers: Combinatorics of flag simplicial 3-polytopes
This paper introduces an algebraic combinatorial approach to simplicial cone decompositions, a key step in solving inhomogeneous linear Diophantine systems and counting lattice points in polytopes. We use constant term manipulation on the…
A polyhedron in Euclidean 3-space is called a regular polyhedron of index 2 if it is combinatorially regular and its geometric symmetry group has index 2 in its combinatorial automorphism group; thus its automorphism group is…
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…
The goal of this paper is to study the link between the topology of the degenerate flag varieties and combinatorics of the Dellac configurations. We define three new classes of algebraic varieties closely related to the degenerate flag…
We describe a construction for d-polytopes generalising the well known stacking operation. The construction is applied to produce 2-simplicial and 2-simple 4-polytopes with g_2=0 on any number of n >= 13 vertices. In particular, this…
We study efficient combinatorial algorithms to produce the Hasse diagram of the poset of bounded faces of an unbounded polyhedron, given vertex-facet incidences. We also discuss the special case of simple polyhedra and present computational…
Refining a basic result of Alexander, we show that two flag simplicial complexes are piecewise linearly homeomorphic if and only if they can be connected by a sequence of flag complexes, each obtained from the previous one by either an edge…
In their paper proving the Hirsch bound for flag normal simplicial complexes (Math. Oper.~Res.~2014) Adiprasito and Benedetti define the notion of~\emph{combinatorial segment}. The study of the maximal length of these objects provides the…
We study a natural generalization of the noncrossing relation between pairs of elements in [n] to k-tuples in [n] that was first considered by Petersen, Pylyavskyy, Speyer (2010). We give an alternative approach to their result that the…
We investigate weighted floating bodies of polytopes. We show that the weighted volume depends on the complete flags of the polytope. This connection is obtained by introducing flag simplices, which translate between the metric and…
We consider the problem of reducing the (semi)total domination number of graph by one by contracting edges. It is known that this can always be done with at most three edge contractions and that deciding whether one edge contraction…
A $3$-polytope is a $3$-connected, planar graph. It is called unigraphic if it does not share its vertex degree sequence with any other $3$-polytope, up to graph isomorphism. The classification of unigraphic $3$-polytopes appears to be a…
Combinatorially and stochastically defined simplicial complexes often have the homotopy type of a wedge of spheres. A prominent conjecture of Kahle quantifies this precisely for the case of random flag complexes. We explore whether such…
Using a ribbon structure of the graph, we construct a dissection of the symmetric edge polytope of a graph into unimodular simplices. Our dissection is shellable, and one can interpret the elements of the resulting $h$-vector via graph…
The graph complexity of a compact 3-manifold is defined as the minimum order among all 4-colored graphs representing it. Exact calculations of graph complexity have been already performed, through tabulations, for closed orientable…
We demonstrate how some problems arising in simplicial quantum gravity can be successfully addressed within the framework of combinatorial group theory. In particular, we argue that the number of simplicial 3-manifolds having a fixed…
We ask which degree sequences admit a unique realisation as a $3$-polytopal graph (polyhedron) on $p$ vertices. We give an exhaustive list of these sequences for the case where one degree equals $p-1$ and exactly two or three of them equal…
Bisztriczky introduced the multiplex as a generalization of the simplex. A polytope is multiplicial if all its faces are multiplexes. In this paper it is proved that the flag vectors of multiplicial polytopes depend only on their face…
When the number of non-triangular faces adjacent to a vertex $v$ is less than or equal to three, the vertex $v$ will be called (\emph{combinatorially}) \emph{rigid}. We study the number of rigid vertices and suggest a conjecture on a…
A polyhedron in Euclidean 3-space is called a regular polyhedron of index 2 if it is combinatorially regular but "fails geometric regularity by a factor of 2"; its combinatorial automorphism group is flag-transitive but its geometric…