On the spectrum of Critical sets in back circulant latin squares
by
Abdollah Khodkar
The University of Queensland
Coauthors: Nicholas J. Cavenagh (The University of Queensland), Diane M. Donovan (The University of Queensland)

A partial latin square P of order n is an n×n array with entries chosen from the set N={0, 1, ..., n-1} in such a way that each element of N occurs at most once in each row and at most once in each column of the array. If all the cells of the array are filled then the partial latin square is termed a latin square. Let B_n denote the back circulant latin square of order n.

Bn={(i, j;i+j(mod  n)) | 0 £ i, j £ n-1}.

In this talk we prove there exists a strong critical set of size m in the back circulant latin square of order n for all (n²-1)/4 £ m £ (n²-n)/2, when n is odd. Moreover, when n is even we prove that there exists a strong critical set of size m in the back circulant latin square of order n for all (n²-n)/2-(n-2) £ m £ (n²-n)/2 and m Î {n²/4, (n²/4)+2, (n²/4)+4, ¼, (n²-n)/2-n}.

Date received: October 19, 2000