Transitive decompositions of graph products: rank 3 grid type

Abstract

A transitive decomposition is a pair (Γ, 𝒫) where Γ is a graph and 𝒫 is a partition of the arc set of Γ, such that there exists a group of automorphisms of Γ which leaves 𝒫 invariant and transitively permutes the parts in 𝒫. This paper concerns transitive decompositions where the group is a primitive rank 3 group of ‘grid’ type. The graphs Γ in this case are either products or Cartesian products of complete graphs. We first give some generic constructions for transitive decompositions of products and Cartesian products of copies of an arbitrary graph, and we then prove (except in a small number of cases) that all transitive decompositions with respect to a rank 3 group of grid type can be characterized using these constructions. Furthermore, the main results of this work provide a new proof and insight into the classification of rank 3 partial linear spaces of product action type studied by Devillers.