On Resolvable Packing RMP(3, 3, V) And Covering RMC(3, 3, V)

Letυandλbe positive integers. A packing (resp. covering) P(K,λ,υ) (resp. C(K,λ,υ)) is an ordered pair (V,β) where V is aυ-set of points, andβis a collection of subsets of V with sizes from K, called blocks, such that each pair of points of V occurs at most (resp. at least)λtimes in the blocks.For any pair e = {x, y} of distinct points , let w(e) be the number of blocks containing e. The leave (resp. excess) of a packing (resp. covering) P(K,λ,υ) (resp. C(K,λ,υ)) is the multigraph spanned by all pairs e of distinct points with multiplicityλ- w(e) (resp. w(e) -λ).A packing (resp. covering) is called resolvable if its block set admits a partition into parallel classes, each parallel class being a partition of the point set V. Denote by RP(K,λ;υ,m) (resp. RC(K,λ;υ,m)) a resolvable packing (resp. covering) P(K,λ,υ) (resp. C(K,λ,υ)) with m parallel classes.Letυ≡k - 1,0 or 1 (mod k). An RMP(k,λ,υ) (resp. RMC(k,,λ,υ)) is a resolvable packing (resp. covering) with maximum (resp. minimum) possible number m(υ) of parallel classes which are mutually distinct, each parallel class consists of [(υ- k + 1)/k] blocks of size k and one block of sizeυ- k[(υ- k + 1)/k], 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, we shall present some new recursive constructions for these designs from some known designs such as frames, large sets of Kirkman triple system, resolvable group divisible designs and incomplete RMPs (RMCs), and we also give some direct constructions for small orders by computer. The existence of an RMP(3, 3,υ) and an RMC(3,3,υ) is proved for any admissibleυexcept for an RMP(3,3,6).