Some partial Latin cubes and their completions

It is well known that all n×n partial Latin squares with at most n−1 entries are completable. Our intent is to extend this well known statement to partial Latin cubes. We show that if an n×n×n partial Latin cube contains at most n−1 entries, no two of which occupy the same row, then the partial Lati...

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Bibliographic Details
Published in:European journal of combinatorics 2011-11, Vol.32 (8), p.1345-1352
Main Authors: Kuhl, Jaromy, Denley, Tristan
Format: Article
Language:English
Subjects:
Combinatorial analysis
Cubes
Extensibility
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Summary:It is well known that all n×n partial Latin squares with at most n−1 entries are completable. Our intent is to extend this well known statement to partial Latin cubes. We show that if an n×n×n partial Latin cube contains at most n−1 entries, no two of which occupy the same row, then the partial Latin cube is completable. Also included in this paper is the problem of completing 2×n×n partial Latin boxes with at most n−1 entries. Given certain sufficient conditions, we show when such partial Latin boxes are completable and then extendable to a deeper Latin box.
ISSN:0195-6698
1095-9971
DOI:10.1016/j.ejc.2011.05.003