| Let λK_v be the complete multigraph with v vertices where any two distinct vertices x and y are joined by λ edges (x,y). Let G be a finite simple graph. A G-design G-GD—λ{v) (G-packing design G-PD—λ(v), G-covering design G~CD—λ(v)) of λK_v, is a pair of (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 exactly (at most, at least) λ blocks of B. A packing (covering) design is said to be maximum (minimum) if no other such packing (covering) design of the same order has more (fewer) blocks. In this paper, maximum packing design and minimum covering design of two graphs with six vertices and nine edges are discussed. By an unified method, we solve this problem for all possible v and λ.In addition, the edge-graceful index-set of W(m, n) is discussed. For three cases among four cases of parity for m and n, we give the corresponding k-edge graceful constructions of W(m,n) and decide the edge-graceful index-set. |
