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Transversals in Latin Squares

A latin square of order n is an n by 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 coding theory through covering radii of sets of permutations.
The survey includes a number of conjectures and open problems.