There are six types of triangles： undirected triangle, cyclic triangle, transitive triangle, mixed-1 triangle, mixed-2 triangle and mixed-3 triangle. The triangle-decompositions for the six types of triangles have already been solved. For the first three types of triangles, their large sets have already been solved, and their overlarge sets have been investigated. In this paper, we establish the spectrum of LTi（v,λ）, OLTi（v）（i = 1, 2）, and give the existence of LT3（v, λ） and OLT3（v, λ） with λ even.
Graph designs for all graphs with six vertices and eight edges are discussed. The existence of these graph designs are completely solved except in two possible cases of order 32.
Let G be a finite simple graph. A G-design (G-packing design, G-covering design) of λKv, denoted by (v, G,λ)-GD ((v, G, λ)-PD, (v, G,λ)-CD), is a pair (X,B) where X is the vertex set of Kv and B is a collection of subgraphs of Kv, called blocks, such that each block is isomorphic to G and any two distinct vertices in Kv 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 has more (fewer) blocks. In this paper, we determine the existence spectrum for the K2,3-designs of λKv,λ> 1, and construct the maximum packing designs and the minimum covering designs of λKv with K2,3 for any integer λ.
