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EXISTENCE OF OPTIMAL STRONG PARTIALLY BALANCED DESIGNS

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Applied Mathematics(A Journal of Chinese Universities) 2007年第2期22卷 169-173页

作者： du beiliang Department of Mathematics Suzhou University Suzhou 215006 China

A strong partially balanced design SPBD（v, b, k; λ,0） whose b is the maximum number of blocks in all SPBD（v, b, k; λ, 0）, as an optimal strong partially balanced design, briefly OSPBD（v, k, λ） is studied. In investigation of authentication codes it has been found that the strong partially balanced design can be used to construct authentication codes. This note investigates the existence of optimal strong partially balanced design OSPBD（v, k, 1） for k = 3 and 4, and shows that there exists an OSPBD（v, k, 1） for any v ≥ k.

关键词： strong partially balanced design pairwise balanced design

P_3-FACTORIZATION OF COMPLETE MULTIPARTITE GRAPHS

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Applied Mathematics(A Journal of Chinese Universities) 1999年第1期14卷 122-124页

作者： du beiliang Depat. of Math Suzhou Univ. Suzhou

In this note it is shown that a necessary and sufficient condition for the existence of a P 3 factorization of complete multipartite graph λK n m is (1) m≥3, (2) mn≡0 (mod 3) and (3) λ(m-1)n≡0 ... 详细信息

关键词： 1991 MR Subject Classification 05B30 05C70

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

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 215006 China Nantong Vocational College Nantong 226007 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 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.

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 1. Vocational College Nantong 226007 China 2. Department of Mathematics Suzhou University Suzhou 215006 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.

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 1. Department of Mathematics Suzhou University 215006 Suzhou China 2. Nantong Vocational College 226007 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.

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 1. Department of Mathematics Suzhou University 215006 Suzhou China 2. Nantong Vocational College 226007 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

SPECTRUM OF DIRECTED KIRKMAN PACKING DESIGNS

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Applied Mathematics(A Journal of Chinese Universities) 2003年第3期18卷 325-331页

作者： Zhang Yan du beiliangDept.of Math.,Suzhou Univ.,Suzhou 215006,China. Dept. of Math. Suzhou Univ. Suzhou China

The problem studied in this article is the directed Kirkman packing,the resolvable directed packing which requires all blocks to be of size three except that,each resolution class should contain either one block of size two(when v≡2(mod 3)) or one block of size four (when v≡1 (mod 3)).A directed Kirkman packing design DKPD(v) is a resolvable directed packing of a v-set by the maximum possible number of resolution classes of this *** article investigates the spectrum of DKPD(v) and it is found that it contains all positive integers v≥3 and v≠5,6.

关键词： packing Kirkman packing Kirkman frame

K_(1,p)~k-FACTORIZATION OF COMPLETE BIPARTITE GRAPHS

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Applied Mathematics(A Journal of Chinese Universities) 2001年第2期16卷 107-110页

作者： du beiliangDept.ofMath.,SuzhouUniv.,Suzhou215006.E-mail:dubl@pub.sz.jsinfo.ne 苏州大学江苏苏州 215006

In this paper, it is shown that a sufficient condition for the existence of a K 1,p k factorization of K m,n , whenever p is a prime number and k is a positive integer, is (1) m≤p kn,(2... 详细信息

In this paper, it is shown that a sufficient condition for the existence of a K 1,p k factorization of K m,n , whenever p is a prime number and k is a positive integer, is (1) m≤p kn,(2) n≤p km,(3)p kn-m≡p km-n ≡0(mod( p 2k -1 )) and (4) (p kn-m)(p km-n) ≡0(mod( p k -1)p k×(p 2k -1)(m+n)) .

关键词： Bipartite graph factor factorization.

某些新的k=6且λ=1的平衡不完全区组设计(英文)

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苏州大学学报(自然科学版) 1986年第4期 465-466页

作者：朱烈杜北梁殷剑兴

Let X be a v-set. Let A be a family of some k-subsets (called bl-ocks) of X. The pair (X, A) is called a balanced incomplete blockdesign, denoted by (u, k, λ) -BIBD, if any two distinct elements ofX are contained in exactly λ blocks of A. Some simple computationshows, for any given integers v, k and λ, a (v, k, λ)-BIBD existsonly if

关键词： BIBD k