# All group-based latin squares possess near transversals

## TL;DR

This paper proves that all group-based Latin squares, specifically those equivalent to finite group Cayley tables, contain near transversals, confirming a longstanding conjecture for this class.

## Contribution

It establishes that the Brualdi-Ryser-Stein conjecture holds for Latin squares derived from finite groups, a significant subclass.

## Key findings

- All group-based Latin squares have near transversals.
- The conjecture is confirmed for Latin squares equivalent to Cayley tables.
- Supports the conjecture's validity in a broad class of Latin squares.

## Abstract

In a latin square of order $n$, a near transversal is a collection of $n-1$ cells which intersects each row, column, and symbol class at most once. A longstanding conjecture of Brualdi, Ryser, and Stein asserts that every latin square possesses a near transversal. We show that this conjecture is true for every latin square that is main class equivalent to the Cayley table of a finite group.