Perfect T(G)-triple Systems For Subgraphs G Of K₅ With Eight Edges Or Less

Let G be a subgraph of K_v.For each edge of G,add a new vertex and replace each edge by a 3-cycle,in which all vertices are distinct each other,then the obtained configuration is called a T(G)-triple.In a T(G)-triple,the edges of G are called interior edges,but the edges of T(G)-G are called exterior edges.An edge-disjoint decomposition of 3K_v into copies of T(G)-triple is called a T(G)-triple system of order v.Further,if all the interior edges in a T(G)-triple system exactly form an edge-disjoint decomposition of K_v,then the T(G)-triple system is said to be perfect.In recent papers by E.J.Billington,C.C.Lindner,S.Kü(?)ük(?)if(?)i and A.Rosa,the existence for perfect T(G)-triple system have been completely solved,where G is any subgraph of K₄.For the star graph K_1,kand any prime power k,Yuanyuan Liu and Qingde Kang have given the spectrums of perfect T(K_1,k)-triple system and perfect T(K_1,2k)-triple system.The same problem for any subgraph G of K₅ with eight edges or less will be discussed in this paper.