Name
Abstract | ||
|---|---|---|
A defining set of a t-(v, k, λ) design is a subcollection of its blocks which is contained in no other t-design with the given parameters, on the same point set. A minimal defining set is a defining set, none of whose proper subcollections is a defining set. The spectrum of minimal defining sets of a design D is the set {|M| | M is a minimal defining set of D}. We show that if a t-(v, k, λ) design D is contained in a design F, then for every minimal defining set d D of D there exists a minimal defining set d F of F such that $${d_D = d_F\cap D}$$. The unique simple design with parameters $${{\left(v,k, {v-2\choose k-2}\right)}}$$is said to be the full design on v elements; it comprises all possible k-tuples on a v set. Every simple t-(v, k, λ) design is contained in a full design, so studying minimal defining sets of full designs gives valuable information about the minimal defining sets of all t-(v, k, λ) designs. This paper studies the minimal defining sets of full designs when t = 2 and k = 3. Several families of non-isomorphic minimal defining sets of these designs are found. For given v, a lower bound on the size of the smallest and an upper bound on the size of the largest minimal defining set are given. The existence of a continuous section of the spectrum comprising approximately v values is shown, where just two values were known previously. |
Year | DOI | Venue |
|---|---|---|
2009 | 10.1007/s00373-010-0882-4 | Graphs and Combinatorics |
Keywords | Field | DocType |
upper bound,spectrum,lower bound | Block size,Discrete mathematics,Combinatorics,Existential quantification,Upper and lower bounds,Point set,Mathematics | Journal |
Volume | Issue | ISSN |
25 | 6 | 0911-0119 |
Citations | PageRank | References |
4 | 0.63 | 1 |
Authors | ||
4 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Diane Donovan | 1 | 72 | 33.88 |
| James G. Lefevre | 2 | 18 | 6.97 |
| Mary Waterhouse | 3 | 11 | 3.38 |
| Emine Şule Yazıcı | 4 | 25 | 7.25 |