A lower bound for the size of a critical set in the back circulant latin square
نویسنده
چکیده
منابع مشابه
On the spectrum of critical sets in latin squares of order 2
Suppose that L is a latin square of order m and P ⊆ L is a partial latin square. If L is the only latin square of order m which contains P , and no proper subset of P has this property, then P is a critical set of L. The critical set spectrum problem is to determine, for a given m, the set of integers t for which there exists a latin square of order m with a critical set of size t. We outline a...
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A critical set in a latin square is a subset of its elements with the following properties: 1) No other latin square exists which also contains that subset. 2) No element may be deleted without destroying property 1. Let scs(n) denote the smallest possible cardinality of a critical set in an n × n latin square. It is conjectured that scs(n) = n/4 , and that only the back-circulant latin square ...
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A general algorithm for finding a minimal critical set for any latin square is presented. By implementing this algorithm, minimal critical sets for all the latin squares of order six have been found. In addition, this algorithm is used to prove that the size of the minimal critical set for a back circulant latin square of order seven is twelve, and for order nine is twenty. These results provid...
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A critical set is a partial latin square that has a unique completion to a latin square of the same order, and is minimal in this property. If P is a critical set in a latin square L, then each element of P must be contained in a latin trade Q in L such that |P ∩ Q| = 1. In the case where each element of P is contained in an intercalate (latin trade of size 4) Q such that |P∩Q| = 1 we say that ...
متن کاملBack circulant Latin squares and the influence of a set
We define the notions of nest and influence of a subset of a critical set of a back circulant latin square, and study their properties. We also show that a secret sharing scheme based on a critical set of a latin square is both compartmentalised, and hierarchical.
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ورودعنوان ژورنال:
- Australasian J. Combinatorics
دوره 36 شماره
صفحات -
تاریخ انتشار 2006