The proof of Ushio's conjecture concerning path factorization of complete bipartite graphs

Let K-m,K-n be a complete bipartite graph with two partite sets having and n vertices, respectively. A P-upsilon-factorization of K-m,K-n is a set of edge-disjoint P-upsilon-factors of K-m,K-n which partition the set of edges of K-m,K-n. When upsilon is an even number, Wang and Ushio gave a necessary and sufficient condition for existence of P-upsilon-factorization of K-m,K-n. When k is an odd number, Ushio in 1993 proposed a conjecture. Very recently, we have proved that Ushio's conjecture is true when upsilon = 4k-1. In this paper we shall show that Ushio Conjecture is true when v = 4k+1, and then Ushio's conjecture is true. That is, we will prove that a necessary and sufficient condition for the existence of a P4k+1-factorization of K-m,K-n is (i) 2km <= (2k+1)n, (ii) 2kn <= (2k+1)m, (iii) m+n equivalent to 0 (mod 4k+1), (iv) (4k+1)mn/[4k(m+n)] is an integer.