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arXiv · 2308.16402

GDD type Spanning Bipartite Block Designs

Shoko Chisaki·Ryoh Fuji-Hara·Nobuko Miyamoto
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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.

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Shoko Chisaki, Ryoh Fuji-Hara, Nobuko Miyamoto. 2023-08-31. GDD type Spanning Bipartite Block Designs. https://arxiv.org/abs/2308.16402

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