Analysis of subsystems with rooks on a chess-board representing a partial Latin square (Part 2.)

TL;DR

This paper explores the representation of partial Latin squares using a 3D chess-board with non-attacking rooks, establishing capacity and balance conditions that determine the completion possibilities of these squares.

Contribution

It introduces a novel chess-board model for partial Latin squares and derives new capacity and balance conditions for their completion, extending previous theoretical results.

Findings

Capacity condition aligns with Cruse's necessary condition.

Partial Latin squares with at most n+1 symbols can be completed if they meet the capacity condition.

Balance condition is necessary for completing a layer in the Latin square.

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. In Latin squares, a subsystem and its most distant mate together have as many rooks as their capacity. That implies a simple capacity condition for the completion of partial Latin squares which is in fact the Cruse's necessary condition for characteristic matrices. Andersen-Hilton proved that, except for certain listed cases, a PLS of order $n$ can be completed if it contains only $n$ symbols. Andersen proved it for $n+1$ symbols, listing the cases to be excluded. Identifying the structures of the chess-board that can be overloaded with $n$ or $n+1$ rooks, it follows that a PLS derived from a chess-board with at most $n+1$ non-attacking rooks can be completed exactly if it satisfies the capacity condition. In a layer of a Latin square, two subsystems of a remote couple are in balance. Thus, a necessary condition for completion of a layer can be formulated, the balance condition. For an LSC, each 1-dimensional subspace of the chess-board contains exactly one rook. Consequently, for the PLSCs derived from partial Latin squares, we examine certain sets of 1-dimensional subspaces because they indicate the number of missing rooks.