Independence Complexes of Stable Kneser Graphs

For integers n\geq 1, k\geq 0, the stable Kneser graph SG_{n,k} (also called the Schrijver graph) has as vertex set the stable n-subsets of [2n+k] and as edges disjoint pairs of n-subsets, where a stable n-subset is one that does not contain any 2-subset of the form {i,i+1} or {1,2n+k}. The stable Kneser graphs have been an interesting object of study since the late 1970's when A. Schrijver determined that they are a vertex critical class of graphs with chromatic number k+2. This article contains a study of the independence complexes of SG_{n,k} for small values of n and k. Our contributions are two-fold: first, we find that the homotopy type of the independence complex of SG_{2,k} is a wedge of spheres of dimension two. Second, we determine the homotopy types of the independence complexes of certain graphs related to SG_{n,2}.