Problems in Latin squares

In mathematics, the theory of Latin squares is an active research area with many open problems. As in other areas of mathematics, such problems are often made public at professional conferences and meetings. Problems posed here appeared in, for instance, the Loops (Prague) conferences and the Milehigh (Denver) conferences.

Open problems

Bounds on maximal number of transversals in a Latin square

A transversal in a Latin square of order n is a set S of n cells such that every row and every column contains exactly one cell of S, and such that the symbols in S form {1,...,n}. Let T(n) be the maximum number of transversals in a Latin square of order n. Estimate T(n).

The minimum number of transversals of a Latin square is also an open problem. H. J. Ryser conjectured (Oberwolfach, 1967) that every Latin square of odd order has one. Closely related is the conjecture, attributed to Richard Brualdi, that every Latin square of order n has a partial transversal of order at least n - 1.

Characterization of Latin subsquares in multiplication tables of Moufang loops

Describe how all Latin subsquares in multiplication tables of Moufang loops arise.

Densest partial Latin squares with Blackburn property

A partial Latin square has Blackburn property if whenever the cells (i,j) and (k,l) are occupied by the same symbol, the opposite corners (i,l) and (k,j) are empty. What is the highest achievable density of filled cells in a partial Latin square with the Blackburn property? In particular, is there some constant c > 0 such that we can always fill at least c n2 cells?

Largest power of 2 dividing the number of Latin squares

Let L_n be the number of Latin squares of order n. What is the largest integer p(n) such that 2^{p(n)} divides L_n? Does p(n) grow quadratically in n?

2 3 4 5 6 7 8 9 10 11
1 1 22 237 26*3*72 210*3*5*1103 217*3*1361291 221*32*5231*3824477 228*32*5*31*37*547135293937 235*34*5*2801*2206499*62368028479
This table suggests that the power of 2 is growing superlinearly. The best current result is that R_n is always divisible by f!, where f is about n/2. See (McKay and Wanless, 2003). Two authors noticed the suspiciously high power of 2 (without being able to shed much light on it): (Alter, 1975), (Mullen, 1978).

See also

References

External links