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Avoiding Arrays of Odd Order by Latin Squares
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
2013 (engelsk)Inngår i: Combinatorics, probability & computing, ISSN 0963-5483, E-ISSN 1469-2163, Vol. 22, nr 2, s. 184-212Artikkel i tidsskrift (Fagfellevurdert) 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.

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2013. Vol. 22, nr 2, s. 184-212
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URN: urn:nbn:se:umu:diva-66768DOI: 10.1017/S0963548312000570ISI: 000314296400002OAI: oai:DiVA.org:umu-66768DiVA, id: diva2:611391
Tilgjengelig fra: 2013-03-15 Laget: 2013-03-05 Sist oppdatert: 2018-06-08bibliografisk kontrollert

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

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