\(P_{5}\)-factorization of complete bipartite graphs (Q2477391)

A \(P_k\)-factor of a graph \(G\) is a spanning subgraph of \(G\) such that every component is a path with \(k\) vertices. A \(P_k\)-factorization of \(G\) is a set of edge-disjoint \(P_k\)-factors of \(G\), which is a partition of the set of edges of \(G\). \textit{K. Ushio} [Discrete Math. 72, 361--366 (1988; Zbl 0667.05045)] gave a necessary and sufficient condition for the existence of \(P_3\)-factorization of a complete bipartite graph \(K_{m,n}\) and raised in [Discrete Math. 116, 299--311 (1993; Zbl 0783.05034)] a conjecture for all \(k\) odd. In this paper the conjecture is proved for \(k=5\).