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Counting Lattice Triangulations
Volker Kaibel, Günter M. Ziegler
We discuss the problem to count, or, more modestly, to estimate the number f(m,n) of unimodular triangulations of the planar grid of size . Among other tools, we employ…
Many Triangulated 3-Spheres
Julian Pfeifle, Günter M. Ziegler
We construct 2^{Ω(n^{5/4})} combinatorial types of triangulated 3-spheres on n vertices. Since by a result of Goodman and Pollack (1986) there are no more than 2^{O(n log n)} combi…
Face numbers of 4-Polytopes and 3-Spheres
Günter M. Ziegler
In this paper, we discuss f- and flag-vectors of 4-dimensional convex polytopes and cellular 3-spheres. We put forward two crucial parameters of fatness and complexity: Fatness F(P…
Topological lower bounds for the chromatic number: A hierarchy
Jiri Matousek, Günter M. Ziegler
This paper is a study of ``topological'' lower bounds for the chromatic number of a graph. Such a lower bound was first introduced by Lovász in 1978, in his famous proof of the \em…
Fat 4-polytopes and fatter 3-spheres
David Eppstein, Greg Kuperberg, Günter M. Ziegler
We introduce the fatness parameter of a 4-dimensional polytope P, defined as ϕ(P)=(f_1+f_2)/(f_0+f_3). It arises in an important open problem in 4-dimensional combinatorial geometr…