On the existence of Κ-homogeneous Latin bitrades

Abstract

Let T be a partial Latin square and L a Latin square such that T ⊆ L.Then T is called a Latin trade, if there exists a partial Latin square T ^* such that T^* ∩ T = ∅ and (L\T) ∪ T^* is a Latin square.We call T^* a disjoint mate of T and the pair (T,T^*) is called a Latin itrade.A Latin bitrade where empty rows and columns are ignored, is called a Κ-homogeneous Latin bitrade, if in each row and each column it contains exactly κ elements, and each element appears exactly k times.The number of filled cells in a Latin trade is referred to as its volume.Following the earlier work on κ-homogeneous Latin bitrades by Cavenagh, Donovan, and Drápal (2003 and 2004) Bean, Bidkhori, Khosravi, and E.S.Mahmoodian (2005) we provethe following re-sults.All κ-homogeneous Latin bitrades of volume km exist, for all odd integers κ and m > k. all even integers κ > 2 and m > min{fc + u, 3k/2}, where u is any odd integer which divides k. all m > k, where 3 < k < 37.