| P_(4k-1)-factorization of complete bipartite graphs |
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| 引用本文： | DU Beiliang & WANG Jian Department of Mathematics,Suzhou University,Suzhou 215006,China, Nantong Vocational College,Nantong 226007,China. P_(4k-1)-factorization of complete bipartite graphs[J]. 中国科学A辑(英文版), 2005, 48(4). DOI: 10.1360/03ys0301 |
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| 作者姓名： | DU Beiliang & WANG Jian Department of Mathematics Suzhou University Suzhou 215006 China Nantong Vocational College Nantong 226007 China |
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| 作者单位： | DU Beiliang & WANG Jian Department of Mathematics，Suzhou University，Suzhou 215006，China; Nantong Vocational College，Nantong 226007，China |
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| 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 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.
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P4k-1-factorization of complete bipartite graphs |
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| DU Beiliang,WANG Jian. P4k-1-factorization of complete bipartite graphs[J]. Science in China(Mathematics), 2005, 48(4). DOI: 10.1360/03ys0301 |
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| Authors: | DU Beiliang WANG Jian |
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| Affiliation: | 1. Department of Mathematics,Suzhou University,Suzhou,215006,China
2. Nantong Vocational College,Nantong,226007,China |
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| Abstract: | 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. |
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| Keywords: | complete bipartite graph factorization Ushio Conjecture |
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