Generalized covering designs and clique coverings

Bailey, Robert F.; Burgess, Andrea C.; Cavers, Michael S.; Meagher, Karen

doi:10.1002/jcd.20288

Cited by 5 publications

(18 citation statements)

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“…In this subsection, we obtain a number of upper bounds on the generalized packing number D λ (v, k, t), particularly when λ = 1. Many of the results are analogous to lower bounds on the sizes of generalized covering designs given in [8]. In many cases, the proofs are sufficiently similar to those in [8] that we refer the reader there for full details.…”

Section: A Few Boundsmentioning

confidence: 67%

“…consisting of the join of the graphs H i such that G v,k has vertex set V = i X i , where |X i | = v i and each X i is the set of vertices of the corresponding H i . Analagous to [8,Theorem 3.5] for generalized covering designs, we have the following result.…”

Section: A Few Boundsmentioning

confidence: 85%

“…We remark that our definition of a generalized packing is identical to Cameron's definition of a generalized t-design, except his definition requires "exactly λ". It is also identical to that given in [8] for generalized covering designs, except that definition requires "at least λ". Clearly, a generalized t-design is simultaneously a generalized packing and a generalized covering design.…”

Section: Definitions and Notationmentioning

confidence: 99%

“…We begin by giving a bound based on the ordinary packing number, which is similar to [8,Corollary 3.10].…”

Section: A Few Boundsmentioning

confidence: 99%

“…, s)) corresponds to orthogonal arrays. Likewise, in [8] one of the motivating examples for generalized covering designs was covering arrays. There is also a "packing" version of these objects, which we define now.…”

Section: A Few Boundsmentioning

confidence: 99%

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Generalized packing designs

Bailey

Burgess

2013

Discrete Mathematics

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Generalized t-designs, which form a common generalization of objects such as t-designs, resolvable designs and orthogonal arrays, were defined by Cameron [P.J. Cameron, A generalisation of t-designs, Discrete Math. 309 (2009), 4835-4842]. In this paper, we define a related class of combinatorial designs which simultaneously generalize packing designs and packing arrays. We describe the sometimes surprising connections which these generalized designs have with various known classes of combinatorial designs, including Howell designs, partial Latin squares and several classes of triple systems, and also concepts such as resolvability and block colouring of ordinary designs and packings, and orthogonal resolutions and colourings. Moreover, we derive bounds on the size of a generalized packing design and construct optimal generalized packings in certain cases. In particular, we provide methods for constructing maximum generalized packings with t = 2 and block size k = 3 or 4.

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Section: A Few Boundsmentioning

confidence: 67%

Section: A Few Boundsmentioning

confidence: 85%

Section: Definitions and Notationmentioning

confidence: 99%

“…We begin by giving a bound based on the ordinary packing number, which is similar to [8,Corollary 3.10].…”

Section: A Few Boundsmentioning

confidence: 99%

Section: A Few Boundsmentioning

confidence: 99%

See 3 more Smart Citations

Generalized packing designs

Bailey

Burgess

2013

Discrete Mathematics

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Asymptotic Bounds for General Covering Designs

Montecalvo

2014

J. Combin. Designs

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Given five positive integers n, p, k, λ, and t where n ≥ k ≥ t and n ≥ p ≥ t, a t-(n, k, p, λ) general covering design is a pair (X, B) where X is a set of n elements (called points) and B a multiset of k-subsets of X (called blocks) such that every p-subset of X intersects at least λ blocks of B in at least t points. In this article we continue the work carried out by Etzion, Wei, and Zhang [Des. Codes Cryptogr. 5 (1995), 217-239] on the asymptotic covering density of general covering designs. We will present combinatorial constructions leading to new upper bounds on the asymptotic covering density of 4-(n, 4, 6, 1) general covering designs and 4-(n, 5, p, 1) general covering designs with 5 ≤ p ≤ 6. The new bound on the asymptotic covering density of 4-(n, 4, 6, 1) general covering designs is equivalent to a new lower bound for the Turán density π (K 4 6 ).

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Optimal partial clique edge covering guided by potential energy minimization

Damaschke

2019

Optim Lett

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For given integers k, n, r we aim at families of k sub-cliques called blocks, of a clique with n vertices, such that every block has r vertices, and the blocks together cover a maximum number of edges. We demonstrate a combinatorial optimization method that generates such optimal partial clique edge coverings. It takes certain packages of columns (corresponding to vertices) in the incidence matrix of the blocks, considers the number of uncovered edges as an energy term that has to be minimized by transforming these packages. As a proof of concept we can completely solve the above maximization problem in the case of k ≤ 4 blocks and obtain optimal coverings for all integers n and r with r /n ≥ 5/9. This generalizes known results for total coverings to partial coverings. The method as such is not restricted to k ≤ 4 blocks, but a challenge for further research (also on total coverings) is to limit the case distinctions when more blocks are involved.

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Generalized covering designs and clique coverings

Abstract: We then obtain a number of bounds on the minimum sizes of these designs, and describe some methods of constructing them, which in some cases we prove are optimal. Many of our results are obtained from an interpretation of these designs in terms of clique coverings of graphs.q

Cited by 5 publications

References 30 publications

Generalized packing designs

Generalized packing designs

Asymptotic Bounds for General Covering Designs

Optimal partial clique edge covering guided by potential energy minimization

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