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- Title
Completing Partial Latin Squares with Blocks of Non-empty Cells.
- Authors
Kuhl, Jaromy; Schroeder, Michael
- Abstract
In this paper we develop two methods for completing partial latin squares and prove the following. Let $$A$$ be a partial latin square of order $$nr$$ in which all non-empty cells occur in at most $$n-1$$ $$r\times r$$ squares. If $$t_1,\ldots , t_m$$ are positive integers for which $$n\geqslant t_1^2+t_2^2+\cdots +t_m^2+1$$ and if $$A$$ is the union of $$m$$ subsquares each with order $$rt_i$$ , then $$A$$ can be completed. We additionally show that if $$n\geqslant r+1$$ and $$A$$ is the union of $$n$$ identical $$r\times r$$ squares with disjoint rows and columns, then $$A$$ can be completed. For smaller values of $$n$$ we show that a completion does not always exist.
- Subjects
SQUARE; MATHEMATICAL proofs; INTEGERS; MATHEMATICAL analysis; NUMERICAL analysis
- Publication
Graphs & Combinatorics, 2016, Vol 32, Issue 1, p241
- ISSN
0911-0119
- Publication type
Article
- DOI
10.1007/s00373-015-1571-0