Paths of homomorphisms from stable Kneser graphs

Abstract

We denote by SG n,k the stable Kneser graph (Schrijver graph) of stable n-subsets of a set of cardinality 2n+k. For k?3 (mod 4) and n?2 we show that there is a component of the ?-colouring graph of SG n,k which is invariant under the action of the automorphism group of SG n,k . We derive that there is a graph G with ?(G)=?(SG n,k ) such that the complex Hom(SG n,k ,G) is non-empty and connected. In particular, for k?3 (mod 4) and n?2 the graph SG n,k is not a test graph.