Book ChapterDOI
Defining Sets for Block Designs: An Update
Anne Penfold Street
- pp 307-320
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This paper deals with the following question: how many, and which, blocks of a design with given parameters must be known before the remaining blocks of the design are uniquely determined.Abstract:
This paper deals with the following question: how many, and which, blocks of a design with given parameters must be known before the remaining blocks of the design are uniquely determined? We survey the theoretical background on such defining sets, some specific results for smallest and other minimal defining sets for small designs and the techniques used in finding them, the few known results on minimal defining sets for infinite classes of designs, and the conjectures on minimal defining sets for some classes of Hadamard designs.read more
Citations
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Journal ArticleDOI
On the forced matching numbers of bipartite graphs
TL;DR: The computational complexity of finding the forcing number of graphs is studied, and some results on the possible values of forcing number for different matchings of the hypercube Q(n) are given.
On the spectrum of the forced matching number of graphs
TL;DR: In this article, the spectrum of possible forced matching numbers for the grids P m × P n is studied. But the problem of finding the smallest forcing number of graphs is NP-complete.
Journal ArticleDOI
Defining sets in vertex colorings of graphs and latin rectangles
TL;DR: The size of smallest defining sets for some classes of graphs are determined and a lower bound is introduced for an arbitrary graph G .
Journal ArticleDOI
The three-way intersection problem for latin squares
TL;DR: The set of integers k for which there exist three latin squares of order n having precisely k cells identical, with their remaining n(2) - k cells different in all three Latin squares, denoted by I-3[n], is determined here for all orders n.
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On the possible volumes of µ-way latin trades
TL;DR: In this paper, the authors show that all μ-way latin trades with sufficiently large volumes exist, and state some theorems on the non-existence of μway Latin trades of certain volumes.
References
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Journal ArticleDOI
On the minimum number of blocks defining a design
TL;DR: A set of blocks which is a subset of a unique t − (v, k, λt) design is said to be a defining set of that design.
Further results on smallest defining sets of well known designs.
TL;DR: Improved lower bounds for the cardinality of a defining set are given for the affine and projective planes of order n.
Minimal defining sets in a family of Steiner triple systems.
TL;DR: It is proven that these sets of blocks in the infinite family of Steiner triple systems isomorphic to the points and lines of the projective geometries over GF(2) are defining sets for the designs and furthermore that they are minimal defining sets.
Journal ArticleDOI
Eight octads suffice
TL;DR: It is possible to produce a set of eight octads which defines the system, whereas any collection of seven octads is a subset of no Steiner system or of more than one.
On small defining sets for some SBIBD(4t-1, 2t-1, t-1)
TL;DR: In this article, the authors conjecture that 2t - 1 specified sets of 2t-1 elements are enough to define an SBIBD(4t -1, 2t − 1, t - 1) when 4t −1 is a prime.
Related Papers (5)
Defining sets in vertex colorings of graphs and latin rectangles
The Size of the Smallest Strong Critical Set in a Latin Square.
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