Transitive path decompositions of Cartesian products of complete graphs

An $H$-decomposition of a graph $\mathrm{\Gamma }$ is a partition of its edge set into subgraphs isomorphic to $H$. A transitive decomposition is a special kind of $H$-decomposition that is highly symmetrical in the sense that the subgraphs (copies of $H$) are preserved and transitively permuted by a group of automorphisms of $\mathrm{\Gamma }$. This paper concerns transitive $H$-decompositions of the graph $K_n\mathrm{\square }{K}_n$ where $H$ is a path. When $n$ is an odd prime, we present a construction for a transitive path decomposition where the paths in the decomposition are considerably large compared to the number of vertices. Our main result supports well-known Gallai’s conjecture and an extended version of Ringel’s conjecture.