Constructions and uses of incomplete pairwise balanced designs

We give explicit constructions for incomplete pairwise balanced designs IPBD\(((v;w),K)\), or, equivalently, edge-decompositions of a difference of two cliques \(K_v \setminus K_w\) into cliques whose sizes belong to the set \(K\). Our constructions produce such designs whenever \(v\) and \(w\) sati...

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Bibliographic Details
Published in:arXiv.org 2018-09
Main Authors: Dukes, Peter J, Lamken, Esther R
Format: Article
Language:English
Subjects:
Arrays
Online Access:Get full text
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Summary:We give explicit constructions for incomplete pairwise balanced designs IPBD\(((v;w),K)\), or, equivalently, edge-decompositions of a difference of two cliques \(K_v \setminus K_w\) into cliques whose sizes belong to the set \(K\). Our constructions produce such designs whenever \(v\) and \(w\) satisfy the usual divisibility conditions, have ratio \(v/w\) bounded away from the smallest value in \(K\) minus one, say \(v/w > k-1+\epsilon\), for \(k =\min K\) and \(\epsilon>0\), and are sufficiently large (depending on \(K\) and \(\epsilon\)). As a consequence, some new results are obtained on many related designs, including class-uniformly resolvable designs, incomplete mutually orthogonal latin squares, and group divisible designs. We also include several other applications that illustrate the power of using IPBDs as `templates'.
ISSN:2331-8422