Decomposing Complete Graphs into Isomorphic Subgraphs with Six Vertices and Seven Edges

Abstract

Let $K_v$ be the complete multigraph with $v$ vertices. Let $G$ be a finite simple graph. A $G$-design of $K_v$, denoted by $G$-GD($v$), is a pair of ($X$, $\mathcal{B}$), where $X$ is the vertex set of $K_v$, and $\mathcal{B}$ is a collection of subgraphs of $K_v$, called blocks, such that each block is isomorphic to $G$ and any two distinct vertices in $K_v$ are joined in exactly one block of $\mathcal{B}$. In this paper, the discussed graphs are sixteen graphs with six vertices and seven edges. We give a unified method for constructing such $G$-designs.