On the existence of resolvable K<sub>4</sub> − e designs

Ge, Gennian; Ling, Alan C. H.

doi:10.1002/jcd.20155

Cited by 9 publications

(7 citation statements)

References 9 publications

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“…Theorem A

(K_{5} ∖ e)

‐GDD of type

g^{n} {(g (n - 1))}^{1}

exists, if

n \geq 3

g (n - 1)

is even and

g n \equiv 0 (mod 3)

, except

(g, n) \in {(2, 3), (2, 6), (6, 3)}

. Proof For each parallel class of a resolvable 3‐GDD of type

g^{n}

(see ), add two infinite points; on each block of the parallel class together with the two infinite points, place a copy of

K_{5} ∖ e

□

Theorem There exists a

(K_{5} ∖ e)

‐GDD of type

1^{n} {(\frac{2 (n - 1)}{5})}^{1}

for each

n \equiv 16 (mod 20)

and

n \geq 16

except possibly for

n = 116, 296

. Proof By , there exists a resolvable

(K_{4} ∖ e)

‐design of order n . For each parallel class, add an infinite point to form copies of

K_{5} ∖ e

.…”

Section: Preliminariesmentioning

confidence: 99%

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A Complete Solution to Spectrum Problem for Five‐Vertex Graphs with Application to Traffic Grooming in Optical Networks

Kolotoğlu

et al. 2014

J of Combinatorial Designs

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A G‐design of order n is a decomposition of the complete graph on n vertices into edge‐disjoint subgraphs isomorphic to G. Grooming uniform all‐to‐all traffic in optical ring networks with grooming ratio C requires the determination of graph decompositions of the complete graph on n vertices into subgraphs each having at most C edges. The drop cost of such a grooming is the total number of vertices of nonzero degree in these subgraphs, and the grooming is optimal when the drop cost is minimum. The existence spectrum problem of G‐designs for five‐vertex graphs is a long standing problem posed by Bermond, Huang, Rosa and Sotteau in 1980, which is closely related to traffic groomings in optical networks. Although considerable progress has been made over the past 30 years, the existence problems for such G‐designs and their related traffic groomings in optical networks are far from complete. In this paper, we first give a complete solution to this spectrum problem for five‐vertex graphs by eliminating all the undetermined possible exceptions. Then, we determine almost completely the minimum drop cost of 8‐groomings for all orders n by reducing the 37 possible exceptions to 8. Finally, we show the minimum possible drop cost of 9‐groomings for all orders n is realizable with 14 exceptions and 12 possible exceptions.

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“…Theorem A

(K_{5} ∖ e)

‐GDD of type

g^{n} {(g (n - 1))}^{1}

exists, if

n \geq 3

g (n - 1)

is even and

g n \equiv 0 (mod 3)

, except

(g, n) \in {(2, 3), (2, 6), (6, 3)}

. Proof For each parallel class of a resolvable 3‐GDD of type

g^{n}

(see ), add two infinite points; on each block of the parallel class together with the two infinite points, place a copy of

K_{5} ∖ e

□

Theorem There exists a

(K_{5} ∖ e)

‐GDD of type

1^{n} {(\frac{2 (n - 1)}{5})}^{1}

for each

n \equiv 16 (mod 20)

and

n \geq 16

except possibly for

n = 116, 296

. Proof By , there exists a resolvable

(K_{4} ∖ e)

‐design of order n . For each parallel class, add an infinite point to form copies of

K_{5} ∖ e

.…”

Section: Preliminariesmentioning

confidence: 99%

“…Proof. By [25], there exists a resolvable (K 4 \e)-design of order n. For each parallel class, add an infinite point to form copies of K 5 \e.…”

Section: Theoremmentioning

confidence: 99%

A Complete Solution to Spectrum Problem for Five‐Vertex Graphs with Application to Traffic Grooming in Optical Networks

Kolotoğlu

et al. 2014

J of Combinatorial Designs

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“…When α = 1, we simply speak of resolvable design and parallel classes. The existence problem of resolvable G-decompositions has been the subject of an extensive research (see [1,4,5,7,8,9,10,11,12,14,15,16,18,19,21,24]). The α-resolvability, with α > 1, has been studied for: G = K 3 by D. Jungnickel, R. C. Mullin, S. A. Vanstone [13], Y. Zhang and B.…”

Section: Introductionmentioning

confidence: 99%

The spectrum of α-resolvable λ-fold (K_4 - e)-designs

et al. 2016

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“…The Kirkman schoolgirl problem is this question for G = K 3 and λ = 1, posed by Kirkman ([9]) in 1847 and solved by Hanani, Ray-Chaudhuri and Wilson ( [7]) in 1969. The question has been studied for: G = K 4 and λ = 1, 3 by Hanani, Ray-Chaudhuri and Wilson ( [7]); G = K 3 and λ = 2 by Hanani ( [6]); G = P 3 and every admissible λ by Horton ([8]); G = P k , k ≥ 4 and every admissible λ by Bermond, Heinrich and Yu ( [1]); G = K 4 − e and λ = 1 by Ge, Ling, Colbourn, Stinson, Whang and Zhu ( [3,5,14]).…”

Section: Introduction and Definitionsmentioning

confidence: 99%

“…• There exists a resolvable (K v , K 4 − e)-design if and only if v ≡ 16 (mod 20) v ≡ 116 (mod 120) ( [3,5,15]).…”

Section: Introduction and Definitionsmentioning

confidence: 99%

Resolvable G-designs of order v and index λ

Gionfriddo,

Faro,

Milici

et al. 2015

Preprint

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On the existence of resolvable K₄ − e designs

Abstract: Abstract:In this article, we settle a problem which originated in [4] regarding the existence of resolvable (K 4 − e)-design. We solve the problem with two possible exceptions.

Cited by 9 publications

References 9 publications

A Complete Solution to Spectrum Problem for Five‐Vertex Graphs with Application to Traffic Grooming in Optical Networks

A Complete Solution to Spectrum Problem for Five‐Vertex Graphs with Application to Traffic Grooming in Optical Networks

The spectrum of α-resolvable λ-fold (K_4 - e)-designs

Resolvable G-designs of order v and index λ

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On the existence of resolvable K4 − e designs

Abstract: Abstract:In this article, we settle a problem which originated in [4] regarding the existence of resolvable (K 4 − e)-design. We solve the problem with two possible exceptions.

Cited by 9 publications

References 9 publications

A Complete Solution to Spectrum Problem for Five‐Vertex Graphs with Application to Traffic Grooming in Optical Networks

A Complete Solution to Spectrum Problem for Five‐Vertex Graphs with Application to Traffic Grooming in Optical Networks

The spectrum of α-resolvable λ-fold (K_4 - e)-designs

Resolvable G-designs of order v and index λ

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Product

Resources

About

On the existence of resolvable K₄ − e designs