Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
9513461 | Discrete Mathematics | 2005 | 7 Pages |
Abstract
A critical set in an nÃn array is a set C of given entries, such that there exists a unique extension of C to an nÃn 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-Ï2n9/6.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.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Discrete Mathematics and Combinatorics
Authors
M. Ghandehari, H. Hatami, E.S. Mahmoodian,