Homogeneous toroidal Latin bitrades

Let T be a partial Latin square. Then T is a Latin trade if there exists a partial Latin square T1, called a trade mate of T, with the properties that (i) a cell is filled in T1 if and only if it is filled in T, (ii) no entry occurs in the same cell in T and T1, (iii) in any given row or column, T and T1 contain the same elements. The pair {T, T1} is called a Latin bitrade. A Latin trade T (and T1) is said to be (r, c, e)- homogeneous if each row contains precisely r entries, each column contains precisely c entries, and each entry occurs precisely e times. An (r, c, e)-homogeneous Latin bitrade can be embedded on the torus only for three parameter sets, namely (r, c, e) = (3, 3, 3), (4, 4, 2) or (6, 3, 2). In this talk I will present classifications for all three cases

Even triangulations and commutative groups

Title: Even triangulations and commutative groups Author: Jan Luber Department: Department of Algebra Supervisor: prof. RNDr. Aleš Drápal, CSc., DSc. Abstract: This thesis takes interest in latin bitrades and triangulations construc- ted from them. Firstly, we introduce needed definitions, properties of the latin bitrades, detailed construction of the triangulation and mainly possibility of em- bedding latin bitrades into abelian groups. These groups are determined by the relations definied on vertices of the triangulation. Then we get concerned with a particular kind of 3-homogeneous latin bitrades which correspond to toroidal tri- angulation whose each vertex has degree six. For these groups we express relation matrix and complement to their torsion ranks. In case of simple triangulations we present explicit description of the groups and with modular arithmetic we get partial description even for more complex triangulations. Keywords: latin bitrade, eulerian triangulation, abelian grou