GDD type Spanning Bipartite Block Designs

Abstract

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 $K_{v_1,v_2}$. A set of subgraphs of $K_{v_1,v_2}$ is constructed from a block set of GDDs. If the GDD satisfies the ${\lambda }_1,{\lambda }_2$ concurrence condition, then the set of subgraphs also satisfies the spanning bipartite block design (SBBD) conditions. We also propose a method to construct SBBD directly from an $(r,\lambda )$-design and a difference matrix over a group. Suppose the $(r,\lambda )$-design consists of $v_2$ points and $v_1$ blocks. When $v_1>>v_2$, we show a method to construct a SBBD with $v_1$ is close to $v_2$ by partitioning the block set.