Holey Mendelsohn Triple System Of Type G~tu~l For G≡0 (Mod 3)

A complete multipartite directed graph DK_{n1,n2,…nh} is a graph with vertexset X = (?)X_i, where X_i (1≤i≤h) are disjoint sets with |X_i|=n_i, such thatany two vertices x and y from different sets X_i and X_j are joined by one arc fromx to y and one arc from y to x. Suppose that DK_{n1,n2,…nh} can be decomposed into 3-circuits. The collection of all these 3-circuits ( called blocks ) is denoted by B.We call (X, B) an Holey Mendelsohn Triple System, denoted by HMTS(v), wherev = (?) is called the order. Each set X_i (1≤i≤h) is called an hole (or a group)and the multiset {n₁, n₂…,n_h} is called the type of the HMTS.In this article, we consider the existence of HMTSs with type g^tu¹. And the main results in this paper is: Let g, t, and u be nonnegative integers and g = 0 (mod 3), u≤g(t-1). Then there exists an HMTS of type g^tu¹.