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Avoiding Arrays of Odd Order by Latin Squares
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
2013 (English)In: Combinatorics, probability & computing, ISSN 0963-5483, E-ISSN 1469-2163, Vol. 22, no 2, 184-212 p.Article in journal (Refereed) Published
Abstract [en]

We prove that there is a constant c such that, for each positive integer k, every (2k + 1) x (2k + 1) array A on the symbols 1, ... , 2k + 1 with at most c(2k + 1) symbols in every cell, and each symbol repeated at most c(2k + 1) times in every row and column is avoidable; that is, there is a (2k + 1) x (2k + 1) Latin square S on the symbols 1, ... , 2k + 1 such that, for each i, j is an element of {1, ... , 2k + 1}, the symbol in position (i, j) of S does not appear in the corresponding cell in Lambda. This settles the last open case of a conjecture by Haggkvist. Using this result, we also show that there is a constant rho, such that, for any positive integer n, if each cell in an n x n array B is assigned a set of m <= rho n symbols, where each set is chosen independently and uniformly at random from {1, ... , n}, then the probability that B is avoidable tends to 1 as n -> infinity.

Place, publisher, year, edition, pages
2013. Vol. 22, no 2, 184-212 p.
National Category
Discrete Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-66768DOI: 10.1017/S0963548312000570ISI: 000314296400002OAI: oai:DiVA.org:umu-66768DiVA: diva2:611391
Available from: 2013-03-15 Created: 2013-03-05 Last updated: 2017-12-06Bibliographically approved

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Andren, Lina J.Casselgren, Carl JohanÖhman, Lars-Daniel

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