Symmetries of the Stable Kneser Graphs

It is well known that the automorphism group of the Kneser graph KGn,k is the symmetric group on n letters. For n ≥ 2k + 1, k ≥ 2, we prove that the automorphism group of the stable Kneser graph SGn,k is the dihedral group of order 2n. Let [n] := [1, 2, 3, . . . , n]. For each n ≥ 2k, n, k ∈ {1, 2, 3, . . .}, the Kneser graph KGn,k has as vertices the k-subsets of [n] with edges defined by disjoint pairs of k-subsets. For the same parameters, the stable Kneser graph SGn,k is the subgraph of KGn,k induced by the stable ksubsets of [n], i.e. those subsets that do not contain any 2-subset of the form {i, i + 1} or {1, n}. The Kneser and stable Kneser graphs are important graphs in algebraic and topological combinatorics. L. Lovász proved in [3] via an ingenious use of the Borsuk-Ulam theorem that χ(KGn,k) = n− 2k + 2, verifying a conjecture due to M. Kneser. Shortly afterwards, A. Schrijver proved in [5], again using the BorsukUlam theorem, that χ(SGn,k) = n− 2k+ 2. Schrijver also proved that the stable Kneser graphs are vertex critical, i.e. the chromatic number of any proper subgraph of SGn,k is strictly less than n − 2k + 2; for this reason, the stable Kneser graphs are also known as the Schrijver graphs. These results sparked a series of dramatic applications of algebraic topology in combinatorics that continues to this day. Despite these general advances, there are many unanswered questions regarding Kneser and stable Kneser graphs. For example, it is well known that for n ≥ 2k+ 1 the automorphism group of the Kneser graph KGn,k is the symmetric group on n letters, with the action induced by the permutation action on [n]; see [2] for a textbook account. The proof of this relies on the Erdős-Ko-Rado theorem characterizing maximal independent sets in KGn,k, where an independent set in a graph is a collection of pairwise non-adjacent vertices. However, the automorphism groups of the stable Kneser graphs are not determined; Date: September 16, 2008. 2000 Mathematics Subject Classification. Primary 05C99, 05E99.