Completing partial Latin squares with two filled rows and three filled columns

TL;DR

This paper develops a framework for completing specific partial Latin squares with fixed filled rows and columns, proving their completability under certain conditions, and advancing understanding of Latin square completion problems.

Contribution

It introduces a new framework based on Kuhl and McGinn's technique for completing partial Latin squares with two filled rows and three filled columns, and proves their completability in specific cases.

Findings

Framework for completing partial Latin squares with fixed filled rows and columns.

Proof that such Latin squares are completable when the intersection forms a Latin rectangle with three symbols.

Extension of the conjecture to a broader class of partial Latin squares.

Abstract

Consider a partial Latin square $P$ where the first two rows and first three columns are completely filled, and every other cell of $P$ is empty. It has been conjectured that all such partial Latin squares of order at least $8$ are completable. Based on a technique by Kuhl and McGinn we describe a framework for completing partial Latin squares in this class. Moreover, we use our method for proving that all partial Latin squares from this family, where the intersection of the nonempty rows and columns form a Latin rectangle with three distinct symbols, is completable.