An analogue of Ryser's Theorem for partial Sudoku squares

TL;DR

This paper extends Ryser's Theorem to partial Sudoku squares, establishing a necessary and sufficient condition for their completion based on Hall's Condition, simplifying the process for certain grid sizes.

Contribution

It introduces Hall's Condition for partial Sudoku squares and proves it as a complete criterion for their completion, generalizing classical results for Latin squares.

Findings

Hall's Condition for partial Sudoku squares is necessary and sufficient for completion.

Any partial (p,q)-Sudoku rectangle with specific size constraints can be completed.

Simplified completion criterion for certain grid sizes where n=pq, p|r, q|s.

Abstract

In 1956 Ryser gave a necessary and sufficient condition for a partial latin rectangle to be completable to a latin square. In 1990 Hilton and Johnson showed that Ryser's condition could be reformulated in terms of Hall's Condition for partial latin squares. Thus Ryser's Theorem can be interpreted as saying that any partial latin rectangle $R$ can be completed if and only if $R$ satisfies Hall's Condition for partial latin squares. We define Hall's Condition for partial Sudoku squares and show that Hall's Condition for partial Sudoku squares gives a criterion for the completion of partial Sudoku rectangles that is both necessary and sufficient. In the particular case where $n=pq$, $p|r$, $q|s$, the result is especially simple, as we show that any $r \times s$ partial $(p,q)$-Sudoku rectangle can be completed (no further condition being necessary).