Transitive path decompositions of Cartesian products of complete graphs

Abstract

An $H$-decomposition of a graph $\hat{I}“$ 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 $\hat{I}“$. This paper concerns transitive $H$-decompositions of the graph $K_n\square K_n$ 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.

15 pages, 4 figures