Neighbour-transitive codes in Johnson graphs

Abstract

The Johnson graph $J(v,k)$ has, as vertices, the $k$-subsets of a $v$-set $\mathcal {V}$ and as edges the pairs of $k$-subsets with intersection of size $k-1$. We introduce the notion of a neighbour-transitive code in $J(v,k)$. This is a proper vertex subset $\Gamma$ such that the subgroup $G$ of graph automorphisms leaving $\Gamma$ invariant is transitive on both the set $\Gamma$ of ‘codewords’ and also the set of ‘neighbours’ of $\Gamma$, which are the non-codewords joined by an edge to some codeword. We classify all examples where the group $G$ is a subgroup of the symmetric group $\mathrm{Sym}\,(\mathcal {V})$ and is intransitive or imprimitive on the underlying $v$-set $\mathcal {V}$. In the remaining case where $G\le \mathrm{Sym}\,(\mathcal {V})$ and $G$ is primitive on $\mathcal {V}$, we prove that, provided distinct codewords are at distance at least $3$, then $G$ is $2$-transitive on $\mathcal {V}$. We examine many of the infinite families of finite $2$-transitive permutation groups and construct surprisingly rich families of examples of neighbour-transitive codes. A major unresolved case remains.