Completing simple partial k-Latin squares

Nicholas Cavenagh, Giovanni Lo Faro, Antoinette Tripodi


We study the completion problem for simple k-Latin rectangles, which are a special case of the generalized latin rectangles studied for which embedding theorems are given by Andersen and Hilton (1980) in “Generalized Latin rectangles II: Embedding”, Discrete Mathematics 31(3). Here an  alternative proof of those theorems are given for k-Latin rectangles in  the “simple” case. More precisely, generalizing two classic results on the  completability of partial Latin squares, we prove the necessary and suffisucient conditions for a completion of a simple m x n k-Latin rectangle to a simple k-Latin square of order and we show that if m ≤ n/2, any simple partial k-Latin square of order embeds in a simple k-Latin square of order n.


Simple k-Factorization; k-Factor; f-Factor; Multi-Latin Square; k- Latin Square; Completion; Generalized Latin Rectangles

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Copyright (c) 2018 Nicholas Cavenagh, Giovanni Lo Faro, Antoinette Tripodi

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