Parity Types, Cycle Structures and Autotopisms of Latin Squares

TL;DR

This paper investigates the parity types of Latin squares, explores their connection to an Alon-Tarsi-like conjecture, and introduces a fast cycle-based algorithm for autotopy group computation.

Contribution

It introduces a new framework for analyzing Latin square parity types, relates them to a conjecture, and develops an efficient autotopy group algorithm.

Findings

Parity types help bound autotopy group sizes

A cycle decomposition algorithm efficiently finds autotopies

Connections to an Alon-Tarsi-like conjecture for odd order squares

Abstract

The parity type of a Latin square is defined in terms of the numbers of even and odd rows and columns. It is related to an Alon-Tarsi-like conjecture that applies to Latin squares of odd order. Parity types are used to derive upper bounds for the size of autotopy groups. A fast algorithm for finding the autotopy group of a Latin square, based on the cycle decomposition of its rows, is presented.