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Application of minimum projection uniformity criterion in complementary designs for q-level factorials
Hong QIN,Zhenghong WANG
Front. Math. China. 2015, 10 (2): 339-350.
https://doi.org/10.1007/s11464-015-0446-2
We study the complementary design problem, which is to express the uniformity pattern of a q-level design in terms of that of its complementary design. Here, a pair of complementary designs form a design in which all the Hamming distances of any two distinct runs are the same, and the uniformity pattern proposed by H. Qin, Z. Wang, and K. Chatterjee [J. Statist. Plann. Inference, 2012, 142: 1170–1177] comes from discrete discrepancy for q-level designs. Based on relationships of the uniformity pattern between a pair of complementary designs, we propose a minimum projection uniformity rule to assess and compare q-level factorials.
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Bondage number of strong product of two paths
Weisheng ZHAO,Heping ZHANG
Front. Math. China. 2015, 10 (2): 435-460.
https://doi.org/10.1007/s11464-014-0391-5
The bondage number b(G) of a graph G is the cardinality of a minimum set of edges whose removal from G results in a graph with a domination number greater than that of G. In this paper, we obtain the exact value of the bondage number of the strong product of two paths. That is, for any two positive integers m≥2 and n≥2, b(Pm?Pn) = 7 - r(m) - r(n) if (r(m), r(n)) = (1, 1) or (3, 3), 6 - r(m) - r(n) otherwise, where r(t) is a function of positive integer t, defined as r(t) = 1 if t ≡ 1 (mod 3), r(t) = 2 if t ≡ 2 (mod 3), and r(t) = 3 if t ≡ 0 (mod 3).
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Almost resolvable maximum packings of complete graphs with 5-cycles
Min ZHOU,Haitao CAO
Front. Math. China. 2015, 10 (2): 461-475.
https://doi.org/10.1007/s11464-015-0425-7
Let X be the vertex set of KnA k-cycle packing of Kn is a triple (X,C,L), where C is a collection of edge disjoint k-cycles of Kn and L is the collection of edges of Kn not belonging to any of the k-cycles in C. A k-cycle packing (X,C,L) is called resolvable if C can be partitioned into almost parallel classes. A resolvable maximum k-cycle packing of Kn, denoted by k-RMCP(n), is a resolvable k-cycle packing of Kn, (X,C,L), in which the number of almost parallel classes is as large as possible. Let D(n, k) denote the number of almost parallel classes in a k-RMCP(n). D(n, k) for k = 3, 4 has been decided. When n ≡ k (mod 2k) and k ≡ 1 (mod 2) or n ≡ 1 (mod 2k) and k ∈{6, 8, 10, 14}∪{m: 5≤m≤49, m ≡ 1 (mod 2)}, D(n, k) also has been decided with few possible exceptions. In this paper, we shall decide D(n, 5) for all values of n≥5.
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14 articles
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