Minimum embedding of P<sub>3</sub>‐designs into (K<sub>4</sub>—e)‐designs

Colbourn, Charles J.; Ling, Alan C. H.; Quattrocchi, Gaetano

doi:10.1002/jcd.10044

Cited by 19 publications

(47 citation statements)

References 12 publications

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“…[5] There exists a G-GDD of type g u if and only if u ≥ 3, g ≡ 0 (mod 5) or u ≡ 0, 1 (mod 5), and (g, u) = (1, 5).…”

Section: Results On G-designsmentioning

confidence: 99%

“…Inflate using weight 2 to form a G-GDD of type 2 2 4 1 and fill the hole to obtain type 2 3 1 2 . For n = 6, take copies of G as follows: [6,2,10,8] [5,13,2,8] [7,4,1,3] [1,12,2,6] [9, 1, 5, 10] [12,9,3,4] [13, 3, 10, 1] [9, 11, 2, 7] [0, 2, 7, 4] [11, 0, 12, 5] [9, 13, 0, 6] [3, 6, 5, 0] [11, 4, 13, 6] [10, 5,7,12] [8,10,0,4] [7, 12, 8, 13] [11, 8, 3, 1]. For n = 8, take copies of G as follows: [2,6,4,17] [ 14, 12, 4, 16] [15, 7, 9, 5] [7, 8, 3, 4] [13, 3, 5, 4] [0, 9, 3, 13] [3, 11, 6, 14] [6,9,12,10] [2,9,16,5] [9, 4, 11, 17] [5, 0, 10, 17] [8,5,12,16] [7, 10, 14, 13] [16, 17, 13, 11] [11, 1, 7, 5] [14, 6, 5, 8] [8, 11, 2, 13].…”

Section: Lemma 22mentioning

confidence: 99%

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Minimum embedding of Steiner triple systems into (K4−e)-designs II

Ling

Colbourn

Quattrocchi

2009

Discrete Mathematics

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A (K 4 − e)-design of order v + w embeds a given Steiner triple system if there is a subset of v points on which the graphs of the design induce the blocks of the original Steiner triple system. It has been established that w ≥ v/3, and that when equality is met, such a minimum embedding of an STS(v) exists, except when v = 15. Equality only holds when v ≡ 15, 27 (mod 30). One natural question is: What is the smallest order w such that some STS(v) can be embedded into a (K 4 − e)-design of order v + w? We solve the problem for 7 of the 10 congruence classes modulo 30.

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“…[5] There exists a G-GDD of type g u if and only if u ≥ 3, g ≡ 0 (mod 5) or u ≡ 0, 1 (mod 5), and (g, u) = (1, 5).…”

Section: Results On G-designsmentioning

confidence: 99%

Section: Lemma 22mentioning

confidence: 99%

Minimum embedding of Steiner triple systems into (K4−e)-designs II

Ling

Colbourn

Quattrocchi

2009

Discrete Mathematics

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“…If the set B of blocks of the graph decomposition consists of b copies of the same graph G, then (X, B) is a G-design of order n. When the two decompositions are a G-design and an H-design, Definition 2.2 coincides with the embedding definition in [28,29]; see also [12,15,30].…”

Section: On (N V ; 3 1)mentioning

confidence: 93%

“…Generalizing the work of Bermond and Coudert [5], we show that the related dynamic grooming problem can be expressed as an optimization problem on a graph decomposition embedding another graph decomposition (see [15,[28][29][30]) and we produce a solution to these problems when C ≤ 3.…”

Section: Introductionmentioning

confidence: 99%

Grooming for two‐period optical networks

2008

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Minimizing the number of add-drop multiplexers (ADMs) in a unidirectional SONET ring can be formulated as a graph decomposition problem. When traffic requirements are uniform and all-to-all, groomings that minimize the number of ADMs (equivalently, the drop cost) have been characterized for grooming ratio at most six. However, when two different traffic requirements are supported, these solutions do not ensure optimality. In two-period optical networks, n vertices are required to support a grooming ratio of C in the first time period, while in the second time period a grooming ratio of C , C < C, is required for v ≤ n vertices. This allows the two-period grooming problem to be expressed as an optimization problem on graph decompositions of K n that embed graph decompositions of K v for v ≤ n. Using this formulation, optimal two-period groomings are found for small grooming ratios using techniques from the theory of graphs and designs.

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“…. This concept has been explored in [7,14], for example. N(n, v; C, C ) denotes an N(n, C) that faithfully embeds an N(v, C ).…”

Section: Theorem 12 ([1])mentioning

confidence: 98%

Lower bounds for two‐period grooming via linear programming duality

2008

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In a problem arising in grooming for two-period optical networks, it is required to decompose the complete graph on n vertices into subgraphs each containing at most C edges, so that the induced subgraphs on a specified set of v ≤ n vertices each contain at most C < C edges. The cost of the grooming is the sum, over all subgraphs, of the number of vertices of nonzero degree in the subgraph. The optimum grooming is the one of lowest cost. An integer linear programming formulation is used to determine precise lower bounds on this minimum cost for all choices of n and v when 1 ≤ C < C ≤ 3. In most cases, this approach determines not only the bound but also the specific structure of any grooming that could realize the bound.

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Minimum embedding of P₃‐designs into (K₄—e)‐designs

Cited by 19 publications

References 12 publications

Minimum embedding of Steiner triple systems into (K4−e)-designs II

Minimum embedding of Steiner triple systems into (K4−e)-designs II

Grooming for two‐period optical networks

Lower bounds for two‐period grooming via linear programming duality

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Minimum embedding of P3‐designs into (K4—e)‐designs

Cited by 19 publications

References 12 publications

Minimum embedding of Steiner triple systems into (K4−e)-designs II

Minimum embedding of Steiner triple systems into (K4−e)-designs II

Grooming for two‐period optical networks

Lower bounds for two‐period grooming via linear programming duality

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Minimum embedding of P₃‐designs into (K₄—e)‐designs