| Summary: | 碩士 === 國立交通大學 === 應用數學系所 === 106 === A Latin square of order n based on an n-set S is an n x n array L such that each element of S occurs in each row and each column of L exactly once. A transversal T of L is a set of n entries one from each column and each row. The number of distinct entries in T is denoted by t_n. If t_n=n, then T is known as a Latin transversal of L.
Given a Latin square of order n, L, it is interesting to know the maximum size of $t_n$ where T is a transversal. Of course, we may list all n! transversals to find the answer. But, if n is getting larger, it is getting more complicate.
A well-known conjecture by Ryser says that t_n=n if n is odd and t_n > or = n-1 if n is even. So far, this conjecture is very far from being settled.
In this thesis, we shall focus on studying the transversals of Latin squares which are obtained from the multiplication table of (abelian) groups. With better structures, as a consequence, we are able to determine whether such Latin squares do have a Latin transversal T. i.e. t_n=n, or not.
|
|---|
