Packings and coverings of λKv by the graphs with seven points, seven edges and one 5-circle

Let lambda K-v be the complete multigraph with v vertices, where any two distinct vertices x and y are joined by A edges {x, y}. Let G be a finite simple graph. A G-packing design (G-covering design) of lambda K-v denoted by (v, G, lambda)-PD ((v, G, lambda)-CD) is a pair (X, B), where X is the vertex set of K-v and 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 at most (at least) lambda blocks of B. A packing (covering) design is said to be maximum (minimum) if no other such packing (covering) design has more (fewer) blocks. There are four graphs with 7 points,7 edges and a 5-circle, denoted by G(i), i = 1, 2, 3, 4. In this paper, we have solved the existence problem of the maximum (v, G(i), lambda)-PD and the minimum (v, G(i), lambda)-CD.