GDD Type Spanning Bipartite Block Designs

There is a one-to-one correspondence between the point set of a group divisible design (GDD) with $v_1$ groups of $v_2$ points and the edge set of a complete bipartite graph $K_{v_1,v_2}$. A block of GDD corresponds to a subgraph of $K_{v_1,v_2}$. We show that the concurrence conditions of two points of GDD can correspond to the edge concurrence conditions of subgraphs of $K_{v_1,v_2}$, which we call a GDD-type spanning bipartite block design (SBBD). We also propose a method to construct SBBD directly from an $\left(r,\lambda \right)$-design and a difference matrix over a group. When an $\left(r,\lambda \right)$-design with $v$ points has $b$ blocks much larger than $v$, a modified method is shown to construct an SBBD with fewer blocks such that $v_1$ is closer to $v_2$ by partitioning the block set of the $\left(r,\lambda \right)$-design.