An analogue of Ryser's Theorem for partial Sudoku squares

Details An analogue of Ryser's Theorem for partial Sudoku squares An analogue of Ryser's Theorem for partial Sudoku squares

2011-07-13

arxiv.org/abs/1107.2634v2

Mathematics

Combinatorics

19 pages, 10 figures

Scientific paper

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).

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