GDDs with two associate classes and with three groups of sizes 1, n and n

Abstract

A group divisible design GDD(v = 1+n+ n, 3, λ1, λ2) is an ordered pair (V, B) where V is an (1 + n + n)-set of symbols and B is a collection of 3-subsets (called blocks) of V satisfying the following properties: the (1 + n + n)-set is divided into 3 groups of sizes 1, n and n; each pair of symbols from the same group occurs in exactly λ1 blocks in B; and each pair of symbols from different groups occurs in exactly λ2 blocks in B. The spectrum of λ1, λ2, denoted by Spec(λ1, λ2), is defined by Spec(λ1, λ2) = {n ∈ N: a GDD(v = 1+n + n, 3, λ1, λ2) exists}. We find the spectrum Spec(λ1, λ2) for all λ1 ≥ λ2.