Transversals in latin squares: a survey

Abstract

A latin square of order n is an n × n array of n symbols in which each symbol occurs exactly once in each row and column. A transversal of such a square is a set of n entries such that no two entries share the same row, column or symbol. Transversals are closely related to the notions of complete mappings and orthomorphisms in (quasi)groups, and are fundamental to the concept of mutually orthogonal latin squares.Here we provide a brief survey of the literature on transversals. We cover (1) existence and enumeration results, (2) generalisations of transversals including partial transversals and plexes, (3) the special case when the latin square is a group table, (4) a connection with covering radii of sets of permutations. The survey includes a number of conjectures and open problems.2000 Mathematics Subject Classifications: 05B15 20N05

On the number of transversals in latin squares

“…The upper bound is provided by (1). The lower bound follows from Theorem 1 and Stirling's formula if n ≡ 1 or 3 mod 6, and it is proved analogously in other cases.…”

“…Email address: vpotapov@math.nsc.ru (Vladimir N. Potapov) 1 The work was funded by the Russian Science Foundation (grant No 14-11-00555).…”

“…n!, where 0 < c 1 < c 2 < 1 and n > 3 is odd, but the known lower bound of T (n) is only exponential (see [1], [2]). Cavenagh and Wanless [3] proved that if n is a sufficiently large odd integer then t(B n ) > (3.246)…”

Latin Squares and Related Structures

“…Further work on transversals has investigated (to name a small number of topics) the largest size of a partial transversal in any latin square of a given order (see [85] for details), the minimum and maximum number of transversals amongst all the latin squares of a given order (again, see [85]), and the possible intersection size of two transversals in a latin squares (in particular, using transversals in the back-circulant latin square B n [30]). …”

“…A transversal of B n is also equivalent to a complete mapping of the cyclic group of order n as well as an orthomorphism of the cyclic group of order n [28] (see also [85]). (Other equivalences can be found in [30].)…”