Packings and Coverings for Four Particular Graphs Each with Six Vertices and Nine Edges $\lambda =1$

Abstract

Let $\lambda K_v$ be the complete multigraph with $v$ vertices. Let $G$ be a finite simple graph. A $G$-design ($G-G{D}_{\lambda })(v)$ ($G$-packing ($G-P{D}_{\lambda })(v)$, $G$-covering ($G-C{D}_{\lambda })(v)$) of $K_v$ is a pair $(X,\mathcal{B})$, 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 exactly (at most, at least) in $\lambda$ blocks. In this paper, we will discuss the maximum packing designs and the minimum covering designs for four particular graphs each with six vertices and nine edges.