Group divisible designs with two associate classes and $({\lambda }_1,{\lambda }_2)$ = (1, 2)

Abstract

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 $v_1,v_2,\dots ,v_g$; 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 $m\le \frac{n}{2}$. 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 $GDD(v=25+33,2,3;1,2)$.