GDD type Spanning Bipartite Block Designs

TL;DR

This paper explores the relationship between group divisible designs and spanning bipartite block designs, proposing methods to construct SBBDs from GDDs and difference matrices, especially when the number of groups is much larger than points.

Contribution

It establishes a correspondence between GDDs and SBBDs and introduces direct construction methods from $(r, \lambda)$-designs and difference matrices.

Findings

Constructed SBBDs from GDDs satisfying concurrence conditions.

Proposed a direct construction method from $(r,\lambda)$-designs.

Developed a partitioning approach for large $v_1$ relative to $v_2$.

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.