P_(4k-1)-factorization of complete bipartite graphs

Let Km,n be a complete bipartite graph with two partite sets having m and n vertices, respectively. A Pv-factorization of Km,n is a set of edge-disjoint pv-factors of Km,n which partition the set of edges of Km,n. When v is an even number, Wang and Ushio gave a necessary and sufficient condition for the existence of Pv-factorization of Km,n.When v is an odd number, Ushio in 1993 proposed a conjecture. However, up to now we only know that Ushio Conjecture is true for v = 3. In this paper we will show that Ushio Conjecture is true when v = 4k - 1. That is, we shall prove that a necessary and sufficient condition for the existence of a P4k-1-factorization of Km,n is (1) (2k - 1)m ≤ 2kn, (2) (2k -1)n≤2km, (3) m n ≡ 0 (mod 4k - 1), (4) (4k -1)mn/2(2k -1)(m n)] is an integer.

P4k-1-factorization of complete bipartite graphs

Let Km,n be a complete bipartite graph with two partite sets having m and n vertices, respectively. A Pv-factorization of Km,n is a set of edge-disjoint Pv-factors of Km,n which partition the set of edges of Km,n. When v is an even number,Wang and Ushio gave a necessary and sufficient condition for the existence of Pv-factorization of Km,n. When v is an odd number, Ushio in 1993 proposed a conjecture. However, up to now we only know that Ushio Conjecture is true for v=3. In this paper wewill show that Ushio Conjecture is true when v=4k-1. That is, we shall prove that a necessary and sufficient condition for the existence of a P4k-1-factorization of Km,n is (1) (2k-1)m≤2kn, (2) (2k-1)n ≤2km, (3) m+n ≡0 (mod 4k-1), (4) (4k-1)mn/2(2k-1)(m+n)] is an integer.integer.