Neighbour-transitive codes in Kneser graphs

Abstract

A code C is a subset of the vertex set of a graph and C is s-neighbour-transitive if its automorphism group $\mathrm{Aut}\left(C\right)$ acts transitively on each of the first $s+1$ parts $C_0,C_1,\dots ,C_s$ of the distance partition $\{C=C_0,C_1,\dots ,C_{\rho }\}$, where ρ is the covering radius of C. While codes have traditionally been studied in the Hamming and Johnson graphs, we consider here codes in the Kneser graphs. Let Ω be the underlying set on which the Kneser graph $K(n,k)$ is defined. Our first main result says that if C is a 2-neighbour-transitive code in $K(n,k)$ such that C has minimum distance at least 5, then $n=2k+1$ (i.e., C is a code in an odd graph) and C lies in a particular infinite family or is one particular sporadic example. We then prove several results when C is a neighbour-transitive code in the Kneser graph $K(n,k)$. First, if $\mathrm{Aut}\left(C\right)$ acts intransitively on Ω we characterise C in terms of certain parameters. We then assume that $\mathrm{Aut}\left(C\right)$ acts transitively on Ω, first proving that if C has minimum distance at least 3 then either $K(n,k)$ is an odd graph or $\mathrm{Aut}\left(C\right)$ has a 2-homogeneous (and hence primitive) action on Ω. We then assume that C is a code in an odd graph and $\mathrm{Aut}\left(C\right)$ acts imprimitively on Ω and characterise C in terms of certain parameters. We give examples in each of these cases and pose several open problems.