Decompositions, Packings And Coverings Of λK_ν Into Graph With Six Vertices And Seven Edges

Let XKV be the complete multigraph with v vertices, where any two distinct vertices x and y are joined by A edges (z, y). Let G be a finite simple graph. A G-design (G-packing design, G-covering design) of XKV, denoted by (v, G,-GD ((v, G, X)-PD; (v, G, A)-C), is a pair of (X, B] where X is the vertex set of Kv and B is a collection of subgraphs of K, called blocks, such that each block is isomorphic to G and any two distinct vertices in Kv are joined in exactly (at most, at least) A blocks of 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, the discussed graphs Gi (1 < i < 8) are eight graphs with six vertices and seven edges. We give an unified method to construct Gi-designs, maximum packings and minimum coverings and completely solve the existence problem for these designs.