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【期刊论文】Kirkman packing designs KPD ({w; s*}, v) and related threshold schemes
杜北梁, H. Cao , , B. Du
H. Gao, B. Du. Discrete Mathematics 281 (2004) 83-95,-0001,():
-1年11月30日
A Kirkman packing design KPD ({w, s*}, v) is a resolvable packing with maximum possible number of parallel classes, each class containing one block of size s and all other blocks of size w. A (t, w)-threshold scheme is a way of distributing partial information (shadows) to w participants, so that any t of them can easily calculate a key, but no subset of fewer than t participants can determine the key. In this paper we improve the existence results on KPD ({3, s*}, v) for s = 4, 5. We also obtain some results on KPD ({4, s*}, v) for s = 5, 6. These results can be used to give some new (2, w)-threshold schemes.
Kirkman packing design, Threshold scheme, 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】Existence of resolvable optimal strong partially balanced designs
杜北梁, Beiliang Du
B. Du. Discrete Applied Mathematics 154 (2006) 930-941,-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). Resolvable strong partially balanced design was first formulated by Wang, Safavi-Naini and Pei [Combinatorial characterization of l-optimal authentication codes with arbitration, J. Combin. Math. Combin. Comput. 37 (2001) 205–224] in investigation of l-optimal authentication codes. This article investigates the existence of resolvable optimal strong partially balanced design ROSPBD(v, 3, 1). We show that there exists an ROSPBD(v, 3, 1) for any v≥3 except v = 6, 12.
Resolvable strong partially balanced design, Kirkman frame
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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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