P_(4k-1)-factorization of bipartite multigraphs  

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

出　　处：《Science China Mathematics》2006年第7期961-970,共10页中国科学：数学（英文版）

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

摘　　要：LetλKm,n be a bipartite multigraph with two partite sets having m and n vertices, respectively. A Pν-factorization ofλKm,n is a set of edge-disjoint Pν-factors ofλKm,n which partition the set of edges ofλKm,n. Whenνis an even number, Ushio, Wang and the second author of the paper gave a necessary and sufficient condition for the existence of a Pν-factorization ofλKm,n. When v is an odd number, we proposed a conjecture. However, up to now we only know that the conjecture is true forν= 3. In this paper we will show that the conjecture is true whenν= 4k-1. That is, we shall prove that a necessary and sufficient condition for the existence of a P4k-1-factorization ofλKm,n is (1) (2k-1)m≤2kn, (2) (2k-1)n≤2km, (3)m + n = 0 (mod 4k-1), (4)λ(4k-1)mn/[2(2k-1)(m + n)] is an integer.LetλKm,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 λKm,n. When v is an odd number, we proposed a conjecture. However, up to now we only know that the conjecture is true for v= 3. In this paper we will show that the 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 λKm,n is (1) (2κ - 1)m ≤ 2kn, (2) (2k - 1)n ≤ 2km, (3) m + n ≡0 (mod 4κ - 1), (4) λ(4κ - 1)mn/[2(2κ - 1)(m + n)] is an integer.

关 键 词：BIPARTITE multigraph  factor  factorization. 

分 类 号：O157.5[理学—数学]