中文摘要：

λKm,n的Pk-分解就是一个（X，β），其中X是λKm,n的顶点集，β是Km,n的子图族，每个子图（称为区组）均同构于Pk，且Km,n中任一边都恰好出现在β的λ个区组中。Ushio在其综述文献中提出了λKm,n的Pk-分解存在性问题的一个猜想。文章证明了该猜想当k=4，5时成立。

英文摘要：

A Pk-decomposition of lKm,n is a pair （X, B） where X is the vertex set of Km,n, and B is a collection of subgraphs of Km,n, called blocks, such that each block is isomorphic to Pk and any two distinct vertices in Km,n are joined in exactly l blocks of B. Ushio proposed a conjecture on the existence of a Pk-decomposition of lKm,n. This paper proves that Ushio conjecture is true when k=4, 5.