Automorphisms of infinite Johnson graph

TL;DR

This paper studies the automorphisms of the infinite Johnson graph, showing that all automorphisms behave regularly on connected components and characterizing regular automorphisms via order-preserving or reversing transformations, with applications to the infinite Kneser graph.

Contribution

It proves that automorphisms are regular on each connected component and characterizes regular automorphisms as order-preserving or reversing bijections.

Findings

Automorphisms are regular on each connected component.

Regular automorphisms are characterized by order-preserving/reversing bijections.

Application to automorphisms of the infinite Kneser graph.

Abstract

We consider the {\it infinite Johnson graph} $J_{\infty}$ whose vertex set consists of all subsets $X\subset {\mathbb N}$ satisfying $|X|=|{\mathbb N}\setminus X|=\infty$ and whose edges are pairs of such subsets $X,Y$ satisfying $|X\setminus Y|=|Y\setminus X|=1$. An automorphism of $J_{\infty}$ is said to be {\it regular} if it is induced by a permutation on $\mathbb{N}$ or it is the composition of the automorphism induced by a permutation on $\mathbb{N}$ and the automorphism $X\to {\mathbb N}\setminus X$. The graph $J_{\infty}$ admits non-regular automorphisms. Our first result states that the restriction of every automorphism of $J_{\infty}$ to any connected component ($J_{\infty}$ is not connected) coincides with the restriction of a regular automorphism. The second result is a characterization of regular automorphisms of $J_{\infty}$ as order preserving and order reversing bijective transformations of the vertex set of $J_{\infty}$ (the vertex set is partially ordered by the inclusion relation). As an application, we describe automorphisms of the associated {\it infinite Kneser graph}.