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On the size of the minimum critical set of a Latin square

Ghandehari, M ; Sharif University of Technology | 2005

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  1. Type of Document: Article
  2. DOI: 10.1016/j.disc.2004.08.026
  3. Publisher: 2005
  4. Abstract:
  5. A critical set in an n×n array is a set C of given entries, such that there exists a unique extension of C to an n×n Latin square and no proper subset of C has this property. For a Latin square L, scs(L) denotes the size of the smallest critical set of L, and scs(n) is the minimum of scs(L) over all Latin squares L of order n. We find an upper bound for the number of partial Latin squares of size k and prove thatn2-(e+o(1))n10/6≤maxscs(L)≤n2- π2n9/6.This improves on a result of Cavenagh (Ph.D. Thesis, The University of Queensland, 2003) and disproves one of his conjectures. Also it improves the previously known lower bound for the size of the largest critical set of any Latin square of order n. © 2005 Elsevier B.V. All rights reserved
  6. Keywords:
  7. Combinatorial mathematics ; Matrix algebra ; Number theory ; Probability ; Theorem proving ; Conjectures ; Critical sets ; Latin squares ; Partial Latin squares ; Set theory
  8. Source: Discrete Mathematics ; Volume 293, Issue 1-3 , 2005 , Pages 121-127 ; 0012365X (ISSN)
  9. URL: https://www.sciencedirect.com/science/article/pii/S0012365X05000579