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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 spectrum of optimal strong partially balanced designs with block size five
杜北梁, Beiliang Du
B. Du. Discrete Mathematics 288 (2004) 19-28,-0001,():
-1年11月30日
We shall refer to 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). The author inpaper (Discrete Math. 279 (2004) 173) investigated the existence of OSPBD(v, 5, 1) and gave the spectra of OSPBD(v, 5, 1) for v ≡ 0, 1, 3 (mod 4). Inthis article we shall investigate the existence of OSPBD(v, 5, 1) and give the spectrum of OSPBD(v, 5, 1) for the remaining case v ≡ 2 (mod 4).
Strong partially balanced design, Incomplete transversal design
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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】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】The existence of orthogonal diagonal Latin squares with subsquares
杜北梁, B. Du
B. Du. Discrete Mathematics 148 (1996) 37-48,-0001,():
-1年11月30日
We prove that there exists a pair of orthogonal diagonal Latin squares of order v with missing subsquares of side n (ODLS (v, n ) for all v≥3n+2 and v-n even. Further, there exists a magic square of order v with missing subsquare of side n (MS (v, n)) for all v≥3n+2 and v-neven.
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