A latin square of order n is an n by n matrix with entries from a set of cardinatlity n such that every row and every column contains each element exactly once.
Euler was the first to work on latin squares in his paper "Sur une nouvelle espace de quarees magique." Since then a tremendous amount of work has been done and a large literature exists. Here I only wish to give a few conjectures which I would like to see proven or a counterexample given. First we need a definition.
Definition: An s-transversal is a set of sn positions on a latin square
meeting each row, column and symbol s times.
Of course is s=1 it is what has always been know as a transversal.
For an interesting paper on transversals, see: I.M. Wanless, A
genralisation of transversals for Latin squares, Elc. J. Combin, 9,
2002, R12, which can be found at www.combinatorics.org
Conjecture 1: (Peter Rodney) Every latin square of even order has a 2-transversal.
This conjecture was made at the Vermont summer conference a few years. The following was made by Jeanette Janssen and me in response.
Conjecture 2: (Janssen/Dougherty) Every s-transversal to a latin square can be partitioned into a k-transversal and an (s-k) transversal if s>2.
Were Conjecture 2 true it would not only give the first conjecture but it would also prove the long standing conjecture that every latin square of odd order has a transversal.
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