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Restricted completion of sparse partial Latin squares

Part of: Designs and configurations Graph theory

Published online by Cambridge University Press:  20 February 2019

Lina J. Andrén ,
Carl Johan Casselgren  and
Klas Markström
Lina J. Andrén
Affiliation:
University Library, Mälardalen University, SE-721 23 Västerås, Sweden
Carl Johan Casselgren*
Affiliation:
Department of Mathematics, Linköping University, SE-581 83 Linköping, Sweden
Klas Markström
Affiliation:
Department of Mathematics and Mathematical Statistics, Umeå University, SE-901 87 Umeå, Sweden
*
*Corresponding author. Email: carl.johan.casselgren@liu.se
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Abstract

An n × n partial Latin square P is called α-dense if each row and column has at most αn non-empty cells and each symbol occurs at most αn times in P. An n × n array A where each cell contains a subset of {1,…, n} is a (βn, βn, βn)-array if each symbol occurs at most βn times in each row and column and each cell contains a set of size at most βn. Combining the notions of completing partial Latin squares and avoiding arrays, we prove that there are constants α, β > 0 such that, for every positive integer n, if P is an α-dense n × n partial Latin square, A is an n × n (βn, βn, βn)-array, and no cell of P contains a symbol that appears in the corresponding cell of A, then there is a completion of P that avoids A; that is, there is a Latin square L that agrees with P on every non-empty cell of P, and, for each i, j satisfying 1 ≤ i, j ≤ n, the symbol in position (i, j) in L does not appear in the corresponding cell of A.

MSC classification

Primary: 05B15: Orthogonal arrays, Latin squares, Room squares
Secondary: 05C15: Coloring of graphs and hypergraphs
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 28 , Issue 5 , September 2019 , pp. 675 - 695
Copyright
© Cambridge University Press 2019 

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Footnotes

†

Part of the work done while the author was a postdoctoral researcher at the Mittag-Leffler Institute. Research supported by a postdoctoral grant from the Mittag-Leffler Institute.

‡

Part of the work done while the author was a postdoctoral researcher at the Mittag-Leffler Institute. Research supported by a postdoctoral grant from the Mittag-Leffler Institute.

§

Part of the work done while the author was visiting the Mittag-Leffler Institute. Research supported by the Mittag-Leffler Institute.

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