Yan Wu

Let $v$ and $\lambda$ be positive integers. A packing (resp. covering) of pairs of points is an ordered pair $({V, \cal B})$ where $V$ is a $v$-set of points, and $\cal B$ is a collection of subsets of $V$, called blocks, such that each pair of points of $V$ occurs at most (resp. at least) $\lambda$ times in the blocks. Let $v\equiv k-1,0$ or $1\ (\mod k)$. An RMP$(k,\lambda,v)$ (resp. RMC$(k,\lambda,v)$) is a resolvable packing (resp. covering) with maximum (resp. minimum) possible number $m$ of parallel classes which are mutually distinct, each parallel class consists of $\lfloor (v-k+1)/k\rfloor$ blocks of size $k$ and one block of size $v-k\lfloor (v-k+1)/k\rfloor$, and its leave (resp. excess) is a simple graph. Such designs were first introduced by Fang and Yin. They have proved that these designs can be used to construct certain uniform designs which has been widely applied in industry, system engineering, pharmaceutics, and natural sciences. In this paper, direct and recursive constructions are discussed for such designs. The existence of an RMP$(3, 3, v)$ and an RMC$(3, 3, v)$ is proved for any admissible $v$.