Name
Abstract | ||
|---|---|---|
It is well known that if n is even, the addition table for the integers modulo n (which we denote by B"n) possesses no transversals. We show that if n is odd, then the number of transversals in B"n is at least exponential in n. Equivalently, for odd n, the number of diagonally cyclic latin squares of order n, the number of complete mappings or orthomorphisms of the cyclic group of order n, the number of magic juggling sequences of period n and the number of placements of n non-attacking semi-queens on an nxn toroidal chessboard are at least exponential in n. For all large n we show that there is a latin square of order n with at least (3.246)^n transversals. We diagnose all possible sizes for the intersection of two transversals in B"n and use this result to complete the spectrum of possible sizes of homogeneous latin bitrades. We also briefly explore potential applications of our results in constructing random mutually orthogonal latin squares. |
Year | DOI | Venue |
|---|---|---|
2010 | 10.1016/j.dam.2009.09.006 | Discrete Applied Mathematics |
Keywords | Field | DocType |
orthomorphism,diagonally cyclic latin square,complete mapping,n transversals,semi-queen,cyclic group,magic juggling sequence,large n,order n,cayley table,possible size,odd n,period n,transversal,random mols,homogeneous latin bitrades,diagonally cyclic,homogeneous latin bitrade,integers modulo n,n non-attacking semi-queens,latin square,spectrum | Diagonal,Integer,Discrete mathematics,Combinatorics,Exponential function,Cyclic group,Modulo,Latin square,Transversal (geometry),Graeco-Latin square,Mathematics | Journal |
Volume | Issue | ISSN |
158 | 2 | Discrete Applied Mathematics |
Citations | PageRank | References |
7 | 0.89 | 13 |
Authors | ||
2 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Nicholas J. Cavenagh | 1 | 92 | 20.89 |
| Ian M. Wanless | 2 | 245 | 38.75 |