P5P5-factorization of complete bipartite graphs

A PkPk-factor of complete bipartite graph Km,nKm,n is a spanning subgraph of Km,nKm,n such that every component is a path of length k  . A PkPk-factorization of Km,nKm,n is a set of edge-disjoint PkPk-factors of Km,nKm,n which is a partition of the set of edges of Km,nKm,n. When k   is an even number, the spectrum problem for a PkPk-factorization of Km,nKm,n has been completely solved. When k   is an odd number, Ushio in 1993 proposed a conjecture. However, up to now we only know that Ushio Conjecture is true for k=3k=3. In this paper we will show that Ushio Conjecture is true when k=5k=5. That is, we shall prove that a necessary and sufficient condition for the existence of a P5P5-factorization of Km,nKm,n is (1) 3n⩾2m3n⩾2m, (2) 3m⩾2n3m⩾2n, (3) m+n≡0m+n≡0 (mod 5), and (4) 5mn/[4(m+n)]5mn/[4(m+n)] is an integer.