A survey on the existence of<i>G</i>‐Designs

Adams, Peter; Bryant, Darryn; Buchanan, Melinda

doi:10.1002/jcd.20170

Cited by 66 publications

(209 citation statements)

References 64 publications

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“…Indeed we settle the existence of G 20 -and G 21 -designs, leaving two possible exceptions (32 and 48) and one possible exception (48), respectively. These results are already included in the published reference [4] and the forthcoming survey [2], which base their reports on the results presented in this paper. …”

Section: Preliminariessupporting

confidence: 70%

Graph designs for the eight-edge five-vertex graphs

Colbourn

Ling

2009

Discrete Mathematics

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a b s t r a c tThe existence of graph designs for the two nonisomorphic graphs on five vertices and eight edges is determined in the case of index one, with three possible exceptions in total. It is established that for the unique graph with vertex sequence (3, 3, 3, 3, 4), a graph design of order n exists exactly when n ≡ 0, 1(mod 16) and n = 16, with the possible exception of n = 48. For the unique graph with vertex sequence (2, 3, 3, 4, 4), a graph design of order n exists exactly when n ≡ 0, 1(mod 16), with the possible exceptions of n ∈ {32, 48}.

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Section: Preliminariessupporting

confidence: 70%

Graph designs for the eight-edge five-vertex graphs

Colbourn

Ling

2009

Discrete Mathematics

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“…Case 2: Suppose that equation (1) has multiple solutions. It is easy to see that the existence of multiple solutions to equation (1) implies that there is a solution with α ≥ k − 2 or a solution with β ≥ k − 1.…”

Section: (V K K − E λ)-Designs With Fewer Than V Blocksmentioning

confidence: 99%

“…We note that G-designs, and (K k − e)-designs in particular, arise in the problem of traffic grooming in optical networks [7,9]. For surveys on G-designs see [1] and [5].…”

Section: Introductionmentioning

confidence: 99%

On the non-existence of pair covering designs with at least as many points as blocks

et al. 2011

Self Cite

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We establish new lower bounds on the pair covering number C λ (v, k) for infinitely many values of v, k and λ, including infinitely many values of v and k for λ = 1. Here, C λ (v, k) denotes the minimum number of k-subsets of a v-set of points such that each pair of points occurs in at least λ of the k-subsets. We use these results to prove simple numerical conditions which are both necessary and sufficient for the existence of (K k − e)-designs with more points than blocks.

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“…The set of all n for which K n admits a G-decomposition is called the spectrum of G. The spectrum has been determined for many classes of graphs, including all graphs on at most 4 vertices [3] and all graphs on 5 vertices (see [4] and [11]). We direct the reader to [2] and [5] for recent surveys on graph decompositions.…”

Section: (G) ∪ V (H) and Edge Set E(g) ∪ E(h)mentioning

confidence: 99%

The spectrum problem for digraphs of order 4 and size 5

Bunge¹,

DeShong²,

El-Zanati³

et al. 2018

OpMath

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Abstract. The paw graph consists of a triangle with a pendant edge attached to one of the three vertices. We obtain a multigraph by adding exactly one repeated edge to the paw. Now, let D be a directed graph obtained by orientating the edges of that multigraph. For 12 of the 18 possibilities for D, we establish necessary and sufficient conditions on n for the existence of a (K * n , D)-design. Partial results are given for the remaining 6 possibilities for D.

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A survey on the existence ofG‐Designs

Cited by 66 publications

References 64 publications

Graph designs for the eight-edge five-vertex graphs

Graph designs for the eight-edge five-vertex graphs

On the non-existence of pair covering designs with at least as many points as blocks

The spectrum problem for digraphs of order 4 and size 5

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