Group divisible designs with two associate classes and \((\lambda_{1}, \lambda_{2})\) = (1, 2)

A group divisible design (GDD) \( (v = v_1 + v_2 + \cdots + v_g, g, k; \lambda_1, \lambda_2) \) is an ordered pair \( (V, \mathcal{B}) \) where \( V \) is a \( v \)-set of symbols and \( \mathcal{B} \) is a collection of \( k \)-subsets (called blocks) of \( V \) satisfying the following properties: the \( v \)-set is divided into \( g \) groups of sizes \( v_1, v_2, \ldots, v_g \); each pair of symbols from the same group occurs in exactly \( \lambda_1 \) blocks in \( \mathcal{B} \); and each pair of symbols from different groups occurs in exactly \( \lambda_2 \) blocks in \( \mathcal{B} \). In this paper we give necessary conditions on \( m \) and \( n \) for the existence of a \( GDD(v = m+n, 2, 3; 1, 2) \), along with sufficient conditions for each \( m \leq \frac{n}{2} \). Furthermore, we introduce some construction techniques to construct some \( GDD(v = m + n, 2, 3; 1, 2) \)s when \( m > \frac{n}{2} \), namely, a \( GDD(v = 9 + 15, 2, 3; 1, 2) \) and a \( GDD(v = 25 + 33, 2, 3; 1, 2) \).