An H-decomposition of a graph \(\Gamma \) 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 \(\Gamma \). This paper concerns transitive H-decompositions of the graph \(K_n \Box K_n\) 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.

图 \（\Gamma \） 的 H 分解是将其边划分为与 H 同构的子图。传递分解是一种特殊的 H 分解，它是高度对称的，因为子图（H 的副本）被保留下来，并被一组 \（\Gamma \） 的自同态传递置换。本文讨论了图 \（K_n \Box K_n\） 的传递 H 分解，其中 H 是一条路径。当 n 是奇数素数时，我们提出了一种传递路径分解的构造，其中分解中的路径与顶点数量相比相当大。我们的主要结果支持著名的 Gallai 猜想和 Ringel 猜想的扩展版本。