Lower bounds for the simplexity of the n-cube

In this paper we prove a new asymptotic lower bound for the minimal number of simplices in simplicial dissections of $n$-dimensional cubes. In particular we show that the number of simplices in dissections of $n$-cubes without additional vertices is at least $(n+1)^{\frac {n-1} 2}$.Comment: 10 page

Hypergraph polynomials and the Bernardi process

Recently O. Bernardi gave a formula for the Tutte polynomial $T(x,y)$ of a graph, based on spanning trees and activities just like the original definition, but using a fixed ribbon structure to order the set of edges in a different way for each tree. The interior polynomial $I$ is a generalization of $T(x,1)$ to hypergraphs. We supply a Bernardi-type description of $I$ using a ribbon structure on the underlying bipartite graph $G$. Our formula works because it is determined by the Ehrhart polynomial of the root polytope of $G$ in the same way as $I$ is. To prove this we interpret the Bernardi process as a way of dissecting the root polytope into simplices, along with a shelling order. We also show that our generalized Bernardi process gives a common extension of bijections (and their inverses) constructed by Baker and Wang between spanning trees and break divisors.Comment: 46 page

Lower Bounds for Simplicial Covers and Triangulations of Cubes

We show that the size of a minimal simplicial cover of a polytope P is a lower bound for the size of a minimal triangulation of P, including ones with extra vertices. We then use this fact to study minimal triangulations of cubes, and we improve lower bounds for covers and triangulations in dimensions 4 through at least 12 (and possibly more dimensions as well). Important ingredients are an analysis of the number of exterior faces that a simplex in the cube can have of a specified dimension and volume, and a characterization of corner simplices in terms of their exterior faces

A combinatorial analysis of topological dissections

AbstractFrom a topological space remove certain subspaces (cuts), leaving connected components (regions). We develop an enumerative theory for the regions in terms of the cuts, with the aid of a theorem on the Möbius algebra of a subset of a distributive lattice. Armed with this theory we study dissections into cellular faces and dissections in the d-sphere. For example, we generalize known enumerations for arrangements of hyperplanes to convex sets and topological arrangements, enumerations for simple arrangements and the Dehn-Sommerville equations for simple polytopes to dissections with general intersection, and enumerations for arrangements of lines and curves and for plane convex sets to dissections by curves of the 2-sphere and planar domains