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Avoiding Arrays of Odd Order by Latin Squares
Umeå University, Sweden .
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
Umeå University, Sweden .
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 andlt;= 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 -andgt; infinity.

Place, publisher, year, edition, pages
Cambridge University Press (CUP) , 2013. Vol. 22, no 2, 184-212 p.
National Category
Natural Sciences
URN: urn:nbn:se:liu:diva-89733DOI: 10.1017/S0963548312000570ISI: 000314296400002OAI: diva2:609384
Available from: 2013-03-05 Created: 2013-03-05 Last updated: 2013-03-05

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Casselgren, Carl Johan
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Mathematics and Applied MathematicsThe Institute of Technology
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