The automorphism group of the bipartite Kneser graph

TL;DR

This paper determines the automorphism group of the bipartite Kneser graph, showing it is isomorphic to the symmetric group on n elements times a cyclic group of order 2, and applies this to Kneser and Johnson graphs.

Contribution

The paper explicitly computes the automorphism group of the bipartite Kneser graph and provides a new proof for the automorphism group of the Kneser graph using this result.

Findings

Aut(H(n, k)) is isomorphic to Sym([n]) × Z_2.

New proof for automorphism group of Kneser graph K(n,k).

Method links automorphism groups of Johnson and Kneser graphs.

Abstract

Let $n$ and $k$ be integers with $n>2k, k\geq1$. We denote by $H(n, k)$ the $bipartite\ Kneser\ graph$, that is, a graph with the family of $k$-subsets and ($n-k$)-subsets of $[n] = \{1, 2, ... , n\}$ as vertices, in which any two vertices are adjacent if and only if one of them is a subset of the other. In this paper, we determine the automorphism group of $H(n, k)$. We show that $Aut(H(n, k))\cong Sym([n]) \times \mathbb{Z}_2$ where $\mathbb{Z}_2$ is the cyclic group of order $2$. Then, as an application of the obtained result, we give a new proof for determining the automorphism group of the Kneser graph $K(n,k)$. In fact we show how to determine the automorphism group of the Kneser graph $K(n,k)$ given the automorphism group of the Johnson graph $J(n,k)$. Note that the known proofs for determining the automorphism groups of Johnson graph $J(n,k)$ and Kneser graph $ K(n,k)$ are independent from each other.