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.

• Proposed: by Ian Wanless at Loops '03, Prague 2003
• Comments: Wanless, McKay and McLeod have bounds of the form cⁿ < T(n) < dⁿ n!, where c > 1 and d is about 0.6. A conjecture by Rivin, Vardi and Zimmermann (Rivin et al., 1994) says that you can place at least exp(c n log n) queens in non-attacking positions on a toroidal chessboard (for some constant c). If true this would imply that T(n) > exp(c n log n). A related question is to estimate the number of transversals in the Cayley tables of cyclic groups of odd order. In other words, how many orthomorphisms do these groups have?

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.

• Proposed: by Aleš Drápal at Loops '03, Prague 2003
• Comments: It is well known that every Latin subsquare in a multiplication table of a group G is of the form aH x Hb, where H is a subgroup of G and a, b are elements of G.

• Proposed: by Ian Wanless at Loops '03, Prague 2003
• Comments: In a paper to appear, Wanless has shown that if c exists then c < 0.463. He also constructed a family of partial Latin squares with the Blackburn property and asymptotic density of at least exp(-d(log n)^1/2) for constant d > 0.

• Proposed: by Ian Wanless at Loops '03, Prague 2003
• Comments: Of course, ${\displaystyle L_{n}=n!(n-1)!R_{n}}$where ${\displaystyle R_{n}}$is the number of reduced Latin squares of order n. This immediately gives a linear number of factors of 2. However, here are the prime factorizations of ${\displaystyle R_{n}}$for n = 2, ...,11:

This table suggests that the power of 2 is growing superlinearly. The best current result is that ${\displaystyle R_{n}}$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).

• Problems in loop theory and quasigroup theory
• Rainbow matching

• Alter, Ronald (1975), "How many latin squares are there?", Amer. Math. Monthly, 82 (6), Mathematical Association of America: 632–634, doi:10.2307/2319697, JSTOR 2319697.
• McKay, Brendan; Wanless, Ian (2005), "On the number of latin squares", Ann. Comb., 9 (3): 335–344, doi:10.1007/s00026-005-0261-7, S2CID 7289396.
• Mullen, Garry (1978), "How many i-j reduced latin squares are there?", Amer. Math. Monthly, 85 (9), Mathematical Association of America: 751–752, doi:10.2307/2321684, JSTOR 2321684.
• Rivin, Igor; Vardi, Ilan; Zimmerman, Paul (1994), "The n-queens problem", Amer. Math. Monthly, 101 (7), Mathematical Association of America: 629–639, doi:10.2307/2974691, JSTOR 2974691.

• Loops '99 conference
• Loops '03 conference
• Loops '07 conference
• Milehigh conference on quasigroups, loops, and nonassociative systems
• LOOPS package for GAP