The automorphism group of the $s$-stable Kneser graphs

Abstract

For $k,s\ge 2$, the $s$-stable Kneser graphs are the graphs with vertex set the $k$-subsets $S$ of $\{1,\dots ,n\}$ such that the circular distance between any two elements in $S$ is at least $s$ and two vertices are adjacent if and only if the corresponding $k$-subset are disjoint. Braun showed that for $n\ge 2k+1$ the automorphism group of the $2$-stable Kneser graphs (Schrijver graphs) is isomorphic to the dihedral group of order $2n$. In this paper we generalize this result by proving that for $s\ge 2$ and $n\ge sk+1$ the automorphism group of the $s$-stable Kneser graphs also is isomorphic to the dihedral group of order $2n$.