Constructions of optimal packing and covering of the complete multi-graph with applications

A G-design (G-packing, G-covering) of lambda K-v is denoted by GD (v, G, lambda) (PD (v, G, lambda), CD (v, G, lambda)). In this paper, we give some construction methods of GD (v, G,,) (PD (v, G, lambda), CD (v, G, lambda)), and determine the packing number of PD (v, K-3(2), lambda) and covering number of CD (v, K-3 (2), lambda) for any lambda >= 1, where K-3(2) denotes the vertex disjoint union of two copies of complete graphs of order 3. Therefore, the problems of G-covering and G-packing of lambda K-v are solved completely, where G is a graph without isolated vertices, vertical bar V(G)vertical bar = 6 and vertical bar E(G)vertical bar <= 6.