Partitioning 3-homogeneous latin bitrades
نویسنده
چکیده
A latin bitrade (T , T⊗) is a pair of partial latin squares that define the difference between two arbitrary latin squares L ⊇ T and L⊗ ⊇ T⊗ of the same order. A 3-homogeneous bitrade (T , T⊗) has three entries in each row, three entries in each column, and each symbol appears three times in T . Cavenagh [2] showed that any 3-homogeneous bitrade may be partitioned into three transversals. In this paper we provide an independent proof of Cavenagh’s result using geometric methods. In doing so we provide a framework for studying bitrades as tessellations in spherical, euclidean or hyperbolic space. Additionally, we show how latin bitrades are related to finite representations of certain triangle groups.
منابع مشابه
Latin bitrades derived from groups
A latin bitrade is a pair of partial latin squares which are disjoint, occupy the same set of non-empty cells, and whose corresponding rows and columns contain the same set of entries. In ([9]) it is shown that a latin bitrade may be thought of as three derangements of the same set, whose product is the identity and whose cycles pairwise have at most one point in common. By letting a group act ...
متن کاملTo the theory of q-ary Steiner and other-type trades
We introduce the concept of a clique bitrade, which generalizes several known types of bitrades, including latin bitrades, Steiner T(k−1, k, v) bitrades, extended 1perfect bitrades. For a distance-regular graph, we show a one-to-one correspondence between the clique bitrades that meet the weight-distribution lower bound on the cardinality and the bipartite isometric subgraphs that are distance-...
متن کاملAn enumeration of spherical latin bitrades
A latin bitrade (T , T⊗) is a pair of partial latin squares which are disjoint, occupy the same set of non-empty cells, and whose corresponding rows and columns contain the same set of entries. A genus may be associated to a latin bitrade by constructing an embedding of the underlying graph in an oriented surface. We report computational enumeration results on the number of spherical (genus 0) ...
متن کاملTransitive latin bitrades
In this note we give two results. First, if a latin bitrade (T , T) is primary, thin, separated, and Aut(T ) acts regularly on T , then (T , T) may be derived from a group-based construction. Second, if a latin bitrade (T , T) has genus 0 then the disjoint mate T is unique and the autotopism group of T ⋄ is equal to the autotopism group of T.
متن کاملOn the number of transversals in Cayley tables of cyclic groups
It is well known that if n is even, the addition table for the integersmodulo n (whichwe denote by Bn) possesses no transversals.We show that ifn is odd, then the number of transversals in Bn is at least exponential in n. Equivalently, for odd n, the number of diagonally cyclic latin squares of order n, the number of completemappings or orthomorphisms of the cyclic group of order n, the number ...
متن کامل