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

作　　者：DU Beiliang & WANG Jian Department of Mathematics, Suzhou University, Suzhou 215006, China Nantong Vocational College, Nantong 226007, China 

出　　处：《Science China Mathematics》2006年第3期289-299,共11页中国科学：数学（英文版）

基　　金：supported by the National Natural Science Foundation of China(Grant No.10571133).

摘　　要：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 existence of Pv-factorization of Km,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 v = 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 Km,n is (i) 2km ≤ (2k + 1)n,(ii) 2kn ≤ (2k + 1)m, (iii) m + n ≡ 0 (mod 4k + 1), (iv) (4k + 1)mn/[4k(m + n)] is an integer.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 existence of Pv-factorization of Km,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 v = 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 Km,n is (i) 2km≤ (2k + 1)n, (ii) 2kn≤ (2k+1)m, (iii) m + n = 0 (mod 4k + 1), (iv) (4k+1)mn/[4k(m + n)] is an integer.

关 键 词：COMPLETE BIPARTITE graph  factorization  Ushio Conjecture. 

分 类 号：O157.5[理学—数学]