Results 1 
3 of
3
Extremal Properties of 0/1Polytopes of Dimension 5
, 1998
"... In this paper we consider polytopes whose vertex coordinates are 0 or 1, so called 0/1polytopes. For the first time we give a complete enumeration of all 0/1polytopes of dimension 5, which enables us to investigate various of their combinatorial extremal properties. For example we show that the ma ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
In this paper we consider polytopes whose vertex coordinates are 0 or 1, so called 0/1polytopes. For the first time we give a complete enumeration of all 0/1polytopes of dimension 5, which enables us to investigate various of their combinatorial extremal properties. For example we show that the maximum number of facets of a fivedimensional 0/1polytope is 40, answering an open question of Ziegler [22]. Based on the complete enumeration for dimension 5 we obtain new results for 2neighbourly 0/1polytopes for higher dimensions. 1 Introduction Among the simplest highdimensional geometric objects is the ddimensional hypercube (dcube). It is a convex polytope that, when having unit edge length, may be expressed as C d = [0; 1] d . Despite of its simple definition, C d has been an object of study from various different points of view. Purely combinatorial properties of C d , mainly involving certain subgraphs formed by its edges and vertices (the latter are just the various d...
Construction of Combinatorial Objects
, 1995
"... Isomorphism problems often can be solved by determining orbits of a group acting on the set of all objects to be classified. The paper centers around algorithms for this topic and shows how to base them on the same idea, the homomorphism principle. Especially it is shown that forming Sims chains, u ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Isomorphism problems often can be solved by determining orbits of a group acting on the set of all objects to be classified. The paper centers around algorithms for this topic and shows how to base them on the same idea, the homomorphism principle. Especially it is shown that forming Sims chains, using an algorithmic version of Burnside's table of marks, computing double coset representatives, and computing Sylow subgroups of automorphism groups can be explained in this way. The exposition is based on graph theoretic concepts to give an easy explanation of data structures for group actions.
COUNTING WITH SYMMETRIES
"... Since I have not found any literature I really like on this subject, these notes are supposed to help you. If you prefer a book I would recommend [1] (which is in the reserve in Millikan) as text. It should be noted that we will consider only finite groups and sets. The essential result (everything ..."
Abstract
 Add to MetaCart
Since I have not found any literature I really like on this subject, these notes are supposed to help you. If you prefer a book I would recommend [1] (which is in the reserve in Millikan) as text. It should be noted that we will consider only finite groups and sets. The essential result (everything we will discuss is either a direct corollary of this result or a slight generalization) is the following result by Frobenius: Theorem (Burnside’s Lemma). Suppose a finite group G acts on a finite set X. Then the number of orbits g∈G Xg, where Xg = {x ∈ X: gx = x}. in X is