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【期刊论文】Kp,q-factorization of complete bipartite graphs
杜北梁, DU Beiliang, WANG Jian
Science in China Ser. A Mathematics 2004, Vol. 47 No. 3, 473-479,-0001,():
-1年11月30日
Let Km,n be a complete bipartite graph with two partite sets having m and n vertices, respectively. A Kp,q factorizeation of Km,n is a set of edge-disjoint Kp,q-factors of Km,n which partition the set of edges of Km,n. When p=1 and q is a prime number, Wang, in his paper “On K1,k-factorizations of a complete bipartite graph” (Discrete Math, 1994, 126: 359-364), investigated the K1,q-factorization of Km,n and gave a sufficient condition for such a factorizeation to exist. In the paper “K1,k-factorizations of complete bipartite graphs” (Discrete Math, 2002, 259: 301-306), Du and Wang extended Wang’s result to the case that q is any positive integer. In this paper, we give a sufficient condition for Km,n to have a Kp,q-factorization. As a special case, it is shown that the Martin’s BAC conjecture is true when p:q=k:(k+1) for any positive integer k.
complete bipartite graph,, factorizeation,, HU BMFS2 scheme.,
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【期刊论文】The proof of Ushio’s conjecture concerning path factorization of complete bipartite graphs
杜北梁, DU Beiliang, WANG Jian
Science in China: Series A Mathematics 2006 Vol. 49, No. 3, 289-299,-0001,():
-1年11月30日
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.
complete bipartite graph,, factorization,, Ushio Conjecture
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【期刊论文】α- Resolvable Group Divisible Designs with Block Size Three
杜北梁, Yan Zhang, Beiliang Du
Published online 16 August 2004 in Wiley InterScience,-0001,():
-1年11月30日
A group divisible design GD (k, λ, t; tu ) isα-resolvable if its blocks can be partitioned into classes such that each point of the design occurs in precisely α blocks in each class. The necessary conditions for the existence of such a design are λt( u – 1) = r ( k – 1) , bk=rtu; k│αtu andα│ r. It is shown in this paper that these conditions are also sufficient when k =3, with some definite exceptions.
group divisible, design, resolvable, frame
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【期刊论文】The existence of three idempotent IMOLS
杜北梁, R. Julian R. Abel, Beiliang Du
R. J. R. Abel. B. Du. Discrete Mathematics 262 (2003) 1-16,-0001,():
-1年11月30日
In this paper it is shown that an idempotent TD (5, m) − TD (5, n) exists whenever the known necessary condition m ≥ 4n+1 is satis3ed, except when (m, n)=(6, 1) and possible when (m, n)=(10, 1). For m < 60 and n ≤ 10, we also indicate where several idempotent TD(k, m)−TD(k, n)’s for k = 6, 7 can be found.
Transversal design, Incomplete transversal design, Idempotent incomplete transversal design, Quasi-difference matrix, Orthogonal latin squares, Incomplete orthogonal latin square
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【期刊论文】Splitting Balanced Incomplete Block Designs with Block Size 3
杜北梁, Beiliang Du
Published online 22 June 2004 in Wiley InterScience,-0001,():
-1年11月30日
Splitting balanced incomplete block designs were first formulated by Ogata, Kurosawa, Stinson, and Saido recently in the investigation of authentication codes. This article investigates the existence of splitting balanced incomplete block designs, i.e., (v, 3k, λ)-splitting BIBDs; we give the spectrum of (v, 3 × 2, λ)-splitting BIBDs. As an application, we obtain an infinite class of 2-splitting A-codes.
splitting balanced incomplete design, k-splitting A-code
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