Carsten Schultz: The equivariant topology of stable Kneser graphs

Kneser graphs were the first examples of graphs whose chromatic numbers could only be determined by topological methods. This is either done by the use of graph complexes, a method introduced by Lovász, or by more direct topological constructions, which were found by Bárány. Schrijver extended Bárány's construction to certain subgraphs of Kneser graphs, the stable Kneser graphs.

We take a new look at the Bárány-Schrijver construction and see that it actually also yields interesting information about graph complexes associated to stable Kneser graphs, unifying the two approaches. We also look at a dual construction which helps us to nicely draw stable Kneser graphs in projective spaces. Both constructions together enable us to answer the question which stable Kneser graphs are test graphs in the sense of Lovász/Babson/Kozlov in most cases.