An oriented triple system on a v-set X is called an HTS (v, λ) if it contains both cyclic and transitive triples such that each ordered pair of distinct elements is contained in exactly λ triples of If all the cyclic and transitive triples from X can be partitioned into U, such that each (X) is an HTS (v,λ), then {(X,)}, is called an LHTS (v,λ). In this paper, the existence spectrum of LHTS(v, 2) is completed, that is 4(v-2)=0 (mod 2), 4 (v-2)≥λ, if λ≠0(mod 3) then v≠2(mod 3) and if λ=1 then v≠3.

Let λkv, be the complete multigraph with vvertices and G a finite simple graph.A G-design ( G-packing design, G-covering design) ofλkv, 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, a maximum (v, G, λ)-PD and a minimum (v, G,λ)-CD are constructed for 3 grraphs of 6 vrertices and 7 edges.