一个六点七边图的填充与覆盖

$\lambda{K_v}$为$\lambda$重$v$点完全图, $G$ 为有限简单图. $\lambda {K_v}$ 的一个 $G$-设计 ( $G$-填充设计, $G$-覆盖设计), 记为 ($v,G,\lambda$)-$GD$(($v,G,\lambda$)-$PD$, ($v,G,\lambda$)-$CD$), 是指一个序偶($X,\calB$)，其中 $X$ 为 ${K_v}$ 的顶点集， $\cal B$ 为 ${K_v}$ 中同构于 $G$的子图的集合, 称为区组集,使得 ${K_v

Packings and Coverings of a Graph with 6 Vertices and 7 Edges

Let $\lambda{K_v}$ be the complete multigraph with $v$ vertices and $G$ a finite simple graph. A $G$-design ($G$-packing design, $G$-covering design) of $\lambda {K_v}$, denoted by ($v,G,\lambda$)-$GD$ (($v,G,\lambda$)-$PD$, ($v,G,\lambda$)-$CD$), is a pair ($X,\cal B$) where $X$ is the vertex set of ${K_v}$ and $\cal B$ 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 (at most, at least) $\lambda$ blocks of ${\cal B}$. A packing (covering) design is said to be maximum (minimum) if no other such packing (covering) design has more (fewer) blocks. In this paper, a simple graph $G$ with 6 vertices and 7 edges is discussed, and the maximum $G$-$PD(v)$ and the minimum $G$-$CD(v)$ are constructed for all orders $v$.