On Resolvable Packing \(RMP(3, 3, v)\) and Covering \(RMC(3, 3, v)\)

Abstract

Let \(v \equiv k-1, 0, \text{ or } 1 \pmod{k}\). An \(\text{RMP}(k, \lambda, v)\) (resp. \(\text{RMC}(k, \lambda, v)\)) is a resolvable packing (resp. covering) with maximum (resp. minimum) possible number \(m(v)\) of parallel classes which are mutually distinct, each parallel class consists of \(\left\lfloor \frac{v – k + 1}{k} \right\rfloor\) blocks of size \(k\) and one block of size \(v – k \left\lfloor \frac{v – k + 1}{k} \right\rfloor\), 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 have been widely applied in industry, system engineering, pharmaceutics, and natural science. In this paper, direct and recursive constructions are discussed for such designs. The existence of an \(\text{RMP}(3, 3, v)\) and an \(\text{RMC}(3, 3, v)\) is proved for any admissible \(v\).