Colouring games based on autotopisms of Latin hyper-rectangles
نویسندگان
چکیده
منابع مشابه
Orthogonal Latin Rectangles
We use a greedy probabilistic method to prove that for every > 0, every m × n Latin rectangle on n symbols has an orthogonal mate, where m = (1− )n. That is, we show the existence of a second latin rectangle such that no pair of the mn cells receives the same pair of symbols in the two rectangles.
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A k X n Latin rectangle is a k X n matrix with entries from {1,2,.. . , n} such that no entry occurs more than once in any row or column. (Thus each row is a permutation of the integers 1,2,..., n.) Let L(k, n) be the number of k x n Latin rectangles. An outstanding problem is to determine the asymptotic value of L(k, n) as n —• oo, with k bounded by a suitable function of n. The first attack o...
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A k × n Latin rectangle is a k × n array of numbers such that (i) each row is a permutation of [n] = {1, 2, . . . , n} and (ii) each column contains distinct entries. If the first row is 12 · · ·n, the Latin rectangle is said to be reduced. Since the number k × n Latin rectangles is clearly n! times the number of reduced k× n Latin rectangles, we shall henceforth consider only reduced Latin rec...
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Symmetries of a partial Latin square are primarily determined by its autotopism group. Analogously to the case of Latin squares, given an isotopism Θ, the cardinality of the set PLSΘ of partial Latin squares which are invariant under Θ only depends on the conjugacy class of the latter, or, equivalently, on its cycle structure. In the current paper, the cycle structures of the set of autotopisms...
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The parity type of a Latin square is defined in terms of the numbers of even and odd rows and columns. It is related to an Alon-Tarsi-like conjecture that applies to Latin squares of odd order. Parity types are used to derive upper bounds for the size of autotopy groups. A new algorithm for finding the autotopy group of a Latin square, based on the cycle decomposition of its rows, is presented,...
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ژورنال
عنوان ژورنال: Quaestiones Mathematicae
سال: 2018
ISSN: 1607-3606,1727-933X
DOI: 10.2989/16073606.2018.1502214