Transversals in Latin Squares

TL;DR

This paper surveys the literature on transversals in Latin squares, discussing existence, enumeration, generalizations, special cases, and open problems related to their combinatorial properties.

Contribution

It provides a comprehensive overview of current research, conjectures, and open problems concerning transversals in Latin squares.

Findings

Summarizes key existence and enumeration results

Discusses generalizations like partial transversals and plexes

Highlights connections with permutation covering radii

Abstract

A latin square of order $n$ is an $n\times n$ 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.