Let λKm,n be a bipartite multigraph with two partite sets having m and n vertices,respectively. A Pv-factorization of λKm,n is a set of edge-disjoint Pv-factors of λKm,n which partition the set of edges of λKm,n. When v is an even number, Ushio, Wang and the second author of the paper gave a necessary and sufficient condition for the existence of a Pv-factorization of λK When v is an odd number, we have proposed a conjecture. Very recently, we have proved that the conjecture is true when v ＝ 4k - 1. In this paper we shall show that the conjecture is true when v ＝ 4k + 1, and then the conjecture is true. That is, we will prove that the necessary and sufficient conditions for the existence of a P4k+1-factorization of λKm,n are (1) 2km ≤ (2k + 1)n, (2) 2kn ≤ (2k + 1)m, (3) m + n ≡0 (mod 4k + 1), (4) λ(4k + 1)mn/[4k(m + n)] is an integer.