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Restricted completion of sparse partial Latin squares.
Combinatorics, Probability and Computing, 1-21. doi:10.1017/S096354831800055X, Cambridge University Press. Abstract An n × n partial Latin square P is called α-dense if each row and column has at most αnnon-emp times in . An × array where each cell contains a subset of {1,…, } is a (, ) -array if each symbol occurs at most times in each row and column and each cell contains a set of size at most . Combining the notions of completing partial Latin squares and avoiding arrays, we prove that there are constants , > 0 such that, for every positive integer , if is an -dense × partial Latin square, is an × -array, and no cell of contains a symbol that appears in the corresponding cell of , then there is a completion of that avoids ; that is, there is a Latin square that agrees with on every non-empty cell of , and, for each , satisfying 1 ≤ , ≤ , the symbol in position (, ) in does not appear in the corresponding cell of .
Triples of Orthogonal Latin and Youden Rectangles For Small Orders
Journal of Combinatorial Designs, Volume 27, Issue 4, p. 229-250, doi.org/10.1002/jcd.21642 Abstract We have performed a complete enumeration of nonisotopic triples of mutually orthogonal Latin rectangle. Here we will present a census of such triples, classified by various properties, including the order of the autotopism group of the triple. As part of this, we have also achieved the first enumeration of pairwise orthogonal triples of Youden rectangles. We have also studied orthogonal triples of rectangles which are formed by extending mutually orthogonal triples with nontrivial autotopisms one row at a time, and requiring that the autotopism group is nontrivial in each step. This class includes a triple coming from the projective plane of order 8. Here we find a remarkably symmetrical pair of triples of rectangles, formed by juxtaposing two selected copies of complete sets of mutually orthogonal Latin squares of order 4.
