Hans Kneser zum Gedenken

Hans Arnold Heilbronn 1908-1975 Hans

Coloring Reduced Kneser Graphs

The vertex set of a Kneser graph KG(m,n) consists of all n-subsets of the set [m] = {0, 1, . . . ,m − 1}. Two vertices are defined to be adjacent if they are disjoint as subsets. A subset of [m] is called 2stable if 2 ≤ |a − b| ≤ m − 2 for any distinct elements a and b in that subset. The reduced Kneser graph KG2(m,n) is the subgraph of KG(m,n) induced by vertices that are 2-stable subsets. We ...

Kneser Representations of Graphs

The Kneser graph Kn:k for positive integers n ≥ k has as its vertex set the k-element subsets of some n-set, with disjoint sets being adjacent. Every finite simple graph can be found as an induced subgraph of some Kneser graph; this can be viewed as a way of representing graphs by labelling their vertices with sets. We explore questions of finding the smallest representation (both in terms of n...

Complete Kneser Transversals

Let k, d, λ > 1 be integers with d > λ. Let m(k, d, λ) be the maximum positive integer n such that every set of n points (not necessarily in general position) in R has the property that the convex hulls of all k-sets have a common transversal (d − λ)-plane. It turns out that m(k, d, λ) is strongly connected with other interesting problems, for instance, the chromatic number of Kneser hypergraph...

Hamiltonian Kneser Graphs

The Kneser graph K (n; k) has as vertices the k-subsets of f1;2;:::;ng. Two vertices are adjacent if the corresponding k-subsets are disjoint. It was recently proved by the rst author 2] that Kneser graphs have Hamilton cycles for n 3k. In this note, we give a short proof for the case when k divides n. x 1. Preliminaries. Suppose that n k 1 are integers and let n] := f1; 2; :::; ng. We denote t...