A new bound on the size of the largest critical set in a Latin square

  • Authors:

  • Richard Bean;E. S. Mahmoodian

  • Affiliations:

  • Department of Mathematics, Centre for Discrete Mathematics and Computing, The University of Queensland, Qld 4072, Australia;Department of Mathematical Sciences, Sharif University of Technology, P.O. Box 11365-9413, Tehran, Islamic Republic of Iran

  • Venue:
  • Discrete Mathematics - Special issue: Combinatorics 2000
  • Year:
  • 2003

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Abstract

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. The cardinality of the largest critical set in any Latin square of order n is denoted by lcs(n). In 1978, Curran and van Rees proved that lcs(n) ≤ n2 - n. Here, we show that lcs(n) ≤ n2 - 3n + 3.