Automorphisms of infinite Johnson 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}.

10 pages

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