Conditions to Extend Partial Latin Rectangles

TL;DR

This paper generalizes Cruse's 1974 conditions for extending partial Latin rectangles to Latin squares, offering alternative proofs and broader implications for combinatorial design theory.

Contribution

It provides new generalizations and consequences of Cruse's theorem through an alternative proof approach, expanding understanding of Latin rectangle extensions.

Findings

Generalized conditions for extending partial Latin rectangles

Alternative proof of Cruse's theorem

Broader implications for Latin square completion

Abstract

In 1974 Allan Cruse provided necessary and sufficient conditions to extend an $r\times s$ partial latin rectangle consisting of $t$ distinct symbols to a latin square of order $n$. Here we provide some generalizations and consequences of this result. Our results are obtained via an alternative proof of Cruse's theorem.