Another proof of Cruse's theorem and a new necessary condition for completion of partial Latin squares (Part 3.)

TL;DR

This paper provides a new proof of Cruse's theorem for partial Latin squares using a 3D chessboard model, extends the theorem to independent compact bricks, and introduces the BUG condition as a necessary criterion for completion.

Contribution

It offers a unified proof approach for classical theorems and extends Cruse's theorem to new structures, introducing the BUG condition for partial Latin square completion.

Findings

Unified proof method for Hall's, Ryser's, and Cruse's theorems.

Extension of Cruse's theorem to independent compact bricks.

Introduction of the BUG condition as a new necessary criterion.

Abstract

A partial Latin square of order $n$ can be represented by a $3$-dimensional chess-board of size $n\times n\times n$ with at most $n^2$ non-attacking rooks. Based on this representation, we apply a uniform method to prove the M. Hall's, Ryser's and Cruse's theorems for completion of partial Latin squares. With the help of this proof, we extend the scope of Cruse's theorem to compact bricks, which appear to be independent of their environment. Without losing any completion you can replace a dot by a rook if the dot must become rook, or you can eliminate the dots that are known not to become rooks. Therefore, we introduce primary and secondary extension procedures that are repeated as many times as possible. If the procedures do not decide whether a PLSC can be completed or not, a new necessary condition for completion can be formulated for the dot structure of the resulting PLSC, the BUG condition.