Embeddings Of Resolvable Group Divisible Designs

The embedding problem is one of the fundamental and important problems in combinatorial design theory. Both foreign and domestic scholars did many important researches in this field and got many beautiful results. In this paper we will investigate the embedding problem for uniform resolvable group divisible designs with block size three. We will get the necessary and sufficient conditions for arbitrary group size m and arbitrary index λ.Let υ and λ be given positive integers, K and M be two sets of positive integers. Let (X, (?),B) be an ordered triple, where X is a υ-set, (?) forms a partition of X. The elements of (?) are called groups, those of B are called blocks. If the following conditions are satisfied:1) |B| ∈ K for each B ∈B,2) |G| ∈ M for each G∈(?),3) |B ∩ G| ≤ I for each B ∈ B and each G ∈(?),4) Any two points of X from different groups are contained exactly in λ blocks, then (X,(?),B) is called a group divisible design, and denoted GD(K,λ, M;υ). When λ = 1, it is simply denoted GD(K, M; υ). If both K and M contain one element, then the GDD is called uniform.Let (X, (?), B) be a GD(K, λ, M;υ), and P(?)B. If P forms a partition of X, then P is called a parallel class. If B can be partitioned into parallel classes, then (X, (?), B) is called resolvable, and denoted RGD(K, λ, M; ν). It is easy to show that if K contains a single element, then the disign is uniform.Let (X,(?),A) be an RGD(K,λ,M;υ), (Y,H,B) be an RGD(K, λ, M;u). If X C Y, (?)(?) H, and any parallel class of A is contained in some parallel class of B, then we say (X,(?),A) is embedded in (Y,H,B).This paper will investigate the embedding problem for resolvable group divisible...