Neighbour-transitive codes in Kneser graphs

TL;DR

This paper studies neighbour-transitive codes within Kneser graphs, characterizing their structure and automorphism group actions, especially focusing on codes with certain minimum distances and symmetries, and identifying specific families and examples.

Contribution

It provides new characterizations of neighbour-transitive codes in Kneser graphs, especially for minimum distances at least 5, and explores automorphism group actions, including intransitive, transitive, and imprimitive cases.

Findings

If minimum distance ≥ 5, then n=2k+1 and code lies in a specific family or is sporadic.

For minimum distance ≥ 3, automorphism group acts 2-homogeneously or the graph is an odd graph.

Examples provided for each case and several open problems posed.

Abstract

A code $C$ is a subset of the vertex set of a graph and $C$ is $s$-neighbour-transitive if its automorphism group ${\rm Aut}(C)$ acts transitively on each of the first $s+1$ parts $C_0,C_1,\ldots,C_s$ of the distance partition $\{C=C_0,C_1,\ldots,C_\rho\}$, where $\rho$ 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 $\Omega$ 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 ${\rm Aut}(C)$ acts intransitively on $\Omega$ we characterise $C$ in terms of certain parameters. We then assume that ${\rm Aut}(C)$ acts transitively on $\Omega$, first proving that if $C$ has minimum distance at least $3$ then either $K(n,k)$ is an odd graph or ${\rm Aut}(C)$ has a $2$-homogeneous (and hence primitive) action on $\Omega$. We then assume that $C$ is a code in an odd graph and ${\rm Aut}(C)$ acts imprimitively on $\Omega$ and characterise $C$ in terms of certain parameters. We give examples in each of these cases and pose several open problems.