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

TL;DR

This paper generalizes the known automorphism group characterization of 2-stable Kneser graphs to s-stable Kneser graphs, showing they also have dihedral automorphism groups under certain conditions.

Contribution

It proves that for all s ≥ 2 and sufficiently large n, the automorphism group of s-stable Kneser graphs is isomorphic to the dihedral group of order 2n, extending previous results.

Findings

Automorphism group of s-stable Kneser graphs is dihedral for n ≥ sk+1.

Generalizes Braun's result from 2-stable to s-stable Kneser graphs.

Provides a unified understanding of symmetries in s-stable Kneser graphs.

Abstract

For $k,s\geq2$, the $s$-stable Kneser graphs are the graphs with vertex set the $k$-subsets $S$ of $\{1,\ldots,n\}$ 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 $n\geq 2k+1$ 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 $s\geq 2$ and $n\geq sk+1$ the automorphism group of the $s$-stable Kneser graphs also is isomorphic to the dihedral group of order $2n$.