Small Partial Latin Squares that Cannot be Embedded in a Cayley Table

TL;DR

This paper investigates the minimal size of partial Latin squares that cannot be embedded into Cayley tables of groups of a given order, providing classifications and addressing related embedding questions.

Contribution

It answers a longstanding question by classifying the smallest non-embeddable partial Latin squares for each group order and explores related embedding problems.

Findings

Identified smallest non-embeddable partial Latin squares for various group orders

Classified minimal examples that cannot be embedded into Cayley tables

Extended results to variants and related embedding questions

Abstract

We answer a question posed by D\'enes and Keedwell that is equivalent to the following. For each order $n$ what is the smallest size of a partial latin square that cannot be embedded into the Cayley table of any group of order $n$? We also solve some variants of this question and in each case classify the smallest examples that cannot be embedded. We close with a question about embedding of diagonal partial latin squares in Cayley tables.