Defining sets in combinatorics: a survey

Donovan, Diane; Mahmoodian, E. S.; Ramsay, Colin M.; Street, Anne

doi:10.1017/cbo9781107359970.006

Cited by 23 publications

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“…This is analogous to the usual definition of defining sets and critical sets of Latin squares and other combinatorial designs .…”

Section: Introductionmentioning

confidence: 95%

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Defining Sets and Critical Sets in (0,1)‐Matrices

Cavenagh

2012

J of Combinatorial Designs

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If D is a partially filled‐in (0, 1)‐matrix with a unique completion to a (0, 1)‐matrix M (with prescribed row and column sums), we say that D is a defining set for M. If the removal of any entry of D destroys this property (i.e. at least two completions become possible), we say that D is a critical set for M. In this note, we show that the complement of a critical set for a (0, 1)‐matrix M is a defining set for M. We also study the possible sizes (number of filled‐in cells) of defining sets for square matrices M with uniform row and column sums, which are also frequency squares. In particular, we show that when the matrix is of even order 2m and the row and column sums are all equal to m, the smallest possible size of a critical set is precisely m2. We give the exact structure of critical sets with this property. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 21: 253–266, 2013

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“…This is analogous to the usual definition of defining sets and critical sets of Latin squares and other combinatorial designs .…”

Section: Introductionmentioning

confidence: 95%

“…Below is an example of a critical set D of size 28 for a (0,1)-matrix M ∈ A( (4,4,4,4,4,4,4,4), (4,4,4,4,4,4,4,4)).…”

Section: Introductionunclassified

Defining Sets and Critical Sets in (0,1)‐Matrices

Cavenagh

2012

J of Combinatorial Designs

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“…Indeed similar notions have been defined and studied independently under various names in different contexts. For example, the term defining set is used for block designs and graph colorings, and the term forcing set, coined by Harary [Har93], is used for other concepts such as perfect matchings, dominating sets, and geodetics (see the survey [DMRS03]). …”

Section: Introductionmentioning

confidence: 99%

Teaching dimension, VC dimension, and critical sets in Latin squares

Hatami

Qian

2018

Journal of Combinatorics

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A critical set in an n × n Latin square is a minimal set of entries that uniquely identifies it among all Latin squares of the same size. It is conjectured by Nelder in 1979, and later independently by Mahmoodian, and Bate and van Rees that the size of the smallest critical set is ⌊n 2 /4⌋. We prove a lower-bound of n 2 /10 4 for sufficiently large n, and thus confirm the quadratic order predicted by the conjecture. This improves a recent lower-bound of Ω(n 3/2 ) due to Cavenagh and Ramadurai.From the point of view of computational learning theory, the size of the smallest critical set corresponds to the minimum teaching dimension of the set of Latin squares. We study two related notions of dimension from learning theory. We prove a lower-bound of n 2 −(e+o(1))n 5/3for both of the VC-dimension and the recursive teaching dimension.

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“…[5] One of the parameters related to chromatic number is defining number. Though we do not study it here but the interested reader can see [1,2,3,4].…”

Section: Introductionmentioning

confidence: 99%