A063437 Cardinality of largest critical set in any Latin square of order n.

0, 1, 3, 7, 11, 18

Offset: 1

Author

Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 24 2001

Keywords

• nonn
• more

Comments

A critical set in an n X n array is a set C of given entries such that there exists a unique extension of C to an n X n Latin square and no proper subset of C has this property.

The next terms satisfy a(7) >= 25, a(8) >= 37, a(9) >= 44, a(10) >= 57. In the reference it is proved that, for all n, a(n) <= n^2 - 3n + 3.

a(9) >= 45. - Richard Bean, May 01 2002

For n sufficiently large (>= 295), a(n) >= (n^2)*(1-(2 + log 2)/log n) + n*(1 + log(8*Pi)/log n) - (log 2)/(log n). Bean and Mahmoodian also show a(n) <= n^2 - 3n + 3. - Jonathan Vos Post, Jan 03 2007

Links

• Richard Bean and E. S. Mahmoodian, A new bound on the size of the largest critical set in a Latin square, arXiv:math/0107159 [math.CO], 2001.
• Richard Bean and Ebadollah S. Mahmoodian, A new bound on the size of the largest critical set in a Latin square, Discrete Math., 267 (2003), 13-21.
• Mahya Ghandehari, Hamed Hatami and Ebadollah S. Mahmoodian, On the size of the minimum critical set of a Latin square, arXiv:math/0701015 [math.CO], 2006.
• Mahya Ghandehari, Hamed Hatami and Ebadollah S. Mahmoodian, On the size of the minimum critical set of a Latin square, Journal of Discrete Mathematics. 293(1-3) (2005) pp. 121-127.
• Hamed Hatami and Ebadollah S. Mahmoodian, A lower bound for the size of the largest critical sets in Latin squares, arXiv:math/0701014 [math.CO], 2006; Bulletin of the Institute of Combinatorics and its Applications (Canada). 38 (2003) pp. 19-22