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

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Science china Mathematics 2005年第4期48卷 539-547页

作者： du beiliang & wang jian Department of Mathematics, Suzhou University, Suzhou 215006, china nantong vocational college, nantong 226007, china Department of Mathematics Suzhou University Suzhou China nantong vocational college Nantong China

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,*** 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.

关键词： complete bipartite graph, factorization, Ushio Conjecture.

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

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Science china Mathematics 2006年第7期49卷 961-970页

作者： wang jian & du beiliang nantong vocational college, nantong 226007, china Department of Mathematics, Suzhou University, Suzhou 215006, china vocational college Nantong China Department of Mathematics Suzhou University Suzhou China

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.

关键词： bipartite multigraph, factor, factorization.

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

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Science china Mathematics 2006年第3期49卷 289-299页

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.

K_(p,q)-factorization of complete bipartite graphs

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Science china Mathematics 2004年第3期47卷 473-479页

作者： du beiliang wang jian Department of Mathematics, Suzhou University, Suzhou 215006, china nantong vocational college, nantong 226007, china Department of Mathematics Suzhou University Suzhou China nantong vocational college Nantong China

Let Km,n be a completebipartite graph with two partite sets having m and n vertices,respectively. A Kp,q-factorization of Km,n is a set ofedge-disjoint Kp,q-factors of Km,n which partition theset of edges of Km,n. When p=1 and q is a prime number,wang, in his paper 'On K1,k-factorizations of a completebipartite graph' (Discrete Math, 1994, 126: 359-364),investigated the K1,q-factorization of Km,n and gave asufficient condition for such a factorization to exist. In the paper'K1,k-factorizations of complete bipartite graphs' (DiscreteMath, 2002, 259: 301-306), du and wang extended wang's resultto the case that q is any positive integer. In this paper, we give a sufficient condition for Km,n to have aKp,q-factorization. As a special case, it is shown that theMartin's BAC conjecture is true when p:q=k:(k+1) for any positiveinteger k.

关键词： complete bipartite graph factorization HUBMFS 2 scheme

Star-factorization of the Complete Bipartite Multigraphs

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Acta Mathematicae Applicatae Sinica 2023年第2期39卷 239-248页

作者： Jing SHI jian wang bei-liang du School of Science Nantong UniversityNantong 226007China Department of Mathematics Suzhou UniversitySuzhou 215006China Mathematics Teaching and Research Office Nantong Vocational CollegeNantong 226007China

LetλK_(m,n)be a complete bipartite multigraph with two partite sets having m and n vertices,respectively.A K_(p,q)-factorization ofλK_(m,n)is a set of K_(p,q)-factors ofλK_(m,n)which partition the set of edges ofλK_(m,n).Whenλ=1,Martin,in[Complete bipartite factorizations by complete bipartite graphs,Discrete Math.,167/168(1997),461–480],gave simple necessary conditions for such a factorization to exist,and conjectured those conditions are always *** this paper,we will study the K_(p,q)-factorization ofλK_(m,n)for p=1,to show that the necessary conditions for such a factorization are always sufficient whenever related parameters are sufficiently large.

关键词： complete bipartite multigraph star factorization