Group-Divisible Designs with Block Size Four and Group-Type gum1 with m as Large or as Small as Possible

Ge, G.; Rees, Rolf S.; Zhu, Lin

doi:10.1006/jcta.2001.3246

Cited by 39 publications

(93 citation statements)

References 6 publications

(1 reference statement)

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“…There exists a 4-GDD of type 6 u m 1 for every u ≥ 4 and m ≡ 0 mod 3 with 0 ≤ m ≤ 3u − 3 except for (u, m) = (4, 0) and except possibly for (u, m) ∈ {(7, 15), (11,21), (11,24), (11,27), (13,27), (13,33), (17,39), (17,42), (19,45), (19,48), (19,51), (23,60), (23, 63)}. We also employ current existence results on 4-RGDDs.…”

Section: Theorem 411 (See [22 Theorem 16])mentioning

confidence: 99%

“…Replace the blocks in the {5, u+1}-GDD by 4-GDDs of types 9 u , 9 u 3 1 , or 9 4 (3i) 1 (from Theorem 4.10) with i ∈ {0, 1, 2, 3, 4} to obtain the 4-GDDs. Here, the input designs that are 4-GDDs of type 9 u 3 1 when u ≡ 3 mod 4 come from [23].…”

Section: Existence Of 4-gdds Of Type 36 U M 1 For Small Values Of Mmentioning

confidence: 99%

“…For other values of m, start from a TD(7, 9) and apply WFC with weight 4 to the points in the first six groups and weight 1 or 4 to the remaining points. The 4-GDD of type 4 6 1 1 is from [30,23]. Proof: Take a 5-GDD of 4 15 and apply WFC with weight 9 to the points in the first 14 groups and weight 0, 3, 6, 9, or 12 to the remaining points.…”

Section: Lemma 415mentioning

confidence: 99%

See 2 more Smart Citations

Traffic Grooming in Unidirectional Wavelength-Division Multiplexed Rings with Grooming RatioC= 6

Bermond¹,

Colbourn²,

Coudert³

et al. 2005

SIAM J. Discrete Math.

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International audienceSONET/WDM networks using wavelength add-drop multiplexing can be constructed using certain graph decompositions used to form a grooming, consisting of unions of primitive rings. The cost of such a decomposition is the sum, over all graphs in the decomposition, of the number of vertices of nonzero degree in the graph. The existence of such decompositions with minimum cost, when every pair of sites employs no more than $\frac{1}{6}$ of the wavelength capacity, is determined with a finite number of possible exceptions. Indeed, when the number N of sites satisfies $N \equiv 1 \pmod{3}$, the determination is complete, and when $N \equiv 2 \pmod{3}$, the only value left undetermined is N = 17. When $N \equiv 0 \pmod{3}$, a finite number of values of N remain, the largest being N = 2580. The techniques developed rely heavily on tools from combinatorial design theory

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Section: Theorem 411 (See [22 Theorem 16])mentioning

confidence: 99%

Section: Existence Of 4-gdds Of Type 36 U M 1 For Small Values Of Mmentioning

confidence: 99%

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Traffic Grooming in Unidirectional Wavelength-Division Multiplexed Rings with Grooming RatioC= 6

Bermond¹,

Colbourn²,

Coudert³

et al. 2005

SIAM J. Discrete Math.

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show abstract

“…Necessary and sufficient conditions for the existence of 3-GDDs of type g u were determined in [10] and for 3-GDDs of type g u m 1 in [2]. The existence of the 4-GDDs that we will be using was determined in [1], [7], [8], [9]. A convenient reference is [6] where the existence of all the GDDs that are used can be verified.…”

Section: Proofmentioning

confidence: 99%

“…Thus the point 0 separates into two vertices. The vertex associated with the cycle (7,9,10,8) is a middle vertex and it is formed by the triples 9, 0, 7 , 10, 0, 9 , (ii) x, y, z ∈ B ⇒ x, zx, yx ∈ B; (iii) every residual vertex has degree at least 6.…”

Section: Propertiesmentioning

confidence: 99%