6,154 research outputs found
More mutually orthogonal latin squares
AbstractWilson's construction for mutually orthogonal Latin squares is generalized. This generalized construction is used to improve known bounds on the function nr (the largest order for which there do not exist r MOLS). In particular we find n7⩽780, n8⩽4738, n9⩽5842, n10⩽7222, n11⩽7478, n12⩽9286, n13⩽9476, n15⩽10632
Mutually orthogonal latin squares with large holes
Two latin squares are orthogonal if, when they are superimposed, every
ordered pair of symbols appears exactly once. This definition extends naturally
to `incomplete' latin squares each having a hole on the same rows, columns, and
symbols. If an incomplete latin square of order has a hole of order ,
then it is an easy observation that . More generally, if a set of
incomplete mutually orthogonal latin squares of order have a common hole of
order , then . In this article, we prove such sets of
incomplete squares exist for all satisfying
Clique decompositions of multipartite graphs and completion of Latin squares
Our main result essentially reduces the problem of finding an
edge-decomposition of a balanced r-partite graph of large minimum degree into
r-cliques to the problem of finding a fractional r-clique decomposition or an
approximate one. Together with very recent results of Bowditch and Dukes as
well as Montgomery on fractional decompositions into triangles and cliques
respectively, this gives the best known bounds on the minimum degree which
ensures an edge-decomposition of an r-partite graph into r-cliques (subject to
trivially necessary divisibility conditions). The case of triangles translates
into the setting of partially completed Latin squares and more generally the
case of r-cliques translates into the setting of partially completed mutually
orthogonal Latin squares.Comment: 40 pages. To appear in Journal of Combinatorial Theory, Series
Permutation polynomials induced from permutations of subfields, and some complete sets of mutually orthogonal latin squares
We present a general technique for obtaining permutation polynomials over a
finite field from permutations of a subfield. By applying this technique to the
simplest classes of permutation polynomials on the subfield, we obtain several
new families of permutation polynomials. Some of these have the additional
property that both f(x) and f(x)+x induce permutations of the field, which has
combinatorial consequences. We use some of our permutation polynomials to
exhibit complete sets of mutually orthogonal latin squares. In addition, we
solve the open problem from a recent paper by Wu and Lin, and we give simpler
proofs of much more general versions of the results in two other recent papers.Comment: 13 pages; many new result
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