Complete T-(k <sub> 1, K </ Sub>) - Triple Systems,

Let G be a subgraph of K_n．The graph obtained from G by replacing each edge with a 3-cyclewhose the third vertex is distinct from other vertices in the configuration is called a T(G)-triple．Anedge-disjoint decomposition of 3K_n into copies{H_i}_i of T(G)is called a T(G)-triple system oforder n．The copy of G in H_i is called T_i,if these{T_i}_i form an edge-disjoint decomposition ofK_n,then the T(G)-triple system is said to be perfect．Recent papers by authors including Billington,Lindner,K(u|¨)(?)(u|¨)k(?)if(?)i and Rosa have completely solved the existence for perfect T(G)-triple system,where G is any subgraph of K₄．The first section of this paper will discuss the same problem for thestar graph K_1,k．Especially,for prime powers k,we have completely solved the existence of perfectT(K_1,k)and perfect T(K_1,2k)．A complete multiple graphs of order v with indexλ,denoted byλK_v,is an undirected graphwith v vertices,where any two distinct vertices are joined byλedges．Let G be a finite simple graph．A graph design G-GD_λ(v)(graph packing G-PD_λ(v),graph covering G-CD_λ(v))ofλK_v 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,suchthat each block is isomorphic to G and any two distinct vertices in K_v are joined in exact(at most,at least)λblocks of B．A graph packing(covering)is said to be maximum(minimum)if no othersuch graph packing(covering)with the same order has more(fewer)blocks．The second section ofthis paper discuss the graph packing and the graph covering of two graphs with six vertices and eightedges．...