P4k-1-factorization of bipartite multigraphs

Let lambda K-m,K-n be a bipartite multigraph with two partite sets having m and n vertices, respectively. A P-upsilon-factorization of lambda K-m,K-n is a set of edge-disjoint P-upsilon-factors of lambda K-m,K-n which partition the set of edges of lambda K-m,K-n. When upsilon is an even number, Ushio, Wang and the second author of the paper gave a necessary and sufficient condition for the existence of a P-upsilon-factorization of lambda K-m,K-n. When upsilon is an odd number, we proposed a conjecture. However, up to now we only know that the conjecture is true for upsilon = 3. In this paper we will show that the conjecture is true when upsilon = 4k - 1. That is, we shall prove that a necessary and sufficient condition for the existence of a P4k-1-factorization of lambda K-m,K-n is (1) (2k - 1)m <= 2kn, (2) (2k - 1)n <= 2km, (3) m + n equivalent to 0 (mod 4k - 1), (4) lambda(4k - 1)mn/[2(2k - 1)(m + n)] is an integer.