Paths of homomorphisms from stable Kneser graphs

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 congruent 3 (mod 4) and n\ge2 we show that there is a component of the \chi-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 \chi(G)=\chi(SG_{n,k}) such that the complex Hom(SG_{n,k}, G) is non-empty and connected. In particular, for k congruent 3 (mod 4) and n\ge2 the graph SG_{n,k} is not a test graph.