In this paper, we first introduce a special structure that allows us to construct a large set of resolvable Mendelsohn triple systems of orders 2q ＋ 2, or LRMTS（2q ＋ 2）, where q = 6t ＋ 5 is a prime power. Using a computer, we find examples of such structure for t C T = {0, 1, 2, 3, 4, 6, 7, 8, 9, 14, 16, 18, 20, 22, 24}. Furthermore, by a method we introduced in [13], large set of resolvable directed triple systems with the same orders are obtained too. Finally, by the tripling construction and product construction for LRMTS and LRDTS introduced in [2, 20, 21], and by the new results for LR-design in [8], we obtain the existence for LRMTS（v）and LRDTS（v）, where v = 12（t ＋ 1） mi≥0（2.7mi＋1）mi≥0（2.13ni＋1）and t∈T,which provides more infinite family for LRMTS and LRDTS of even orders.
A maximum （v, G, λ）-PD and a minimum （v, G, λ）-CD axe studied for 2 graphs of 6 vertices and 7 edges. By means of ＂difference method＂ and ＂holey graph design＂, we obtain the result： there exists a （v, Gi, λ）-OPD （OCD） for v ≡ 2, 3, 4, 5, 6 （mod 7）, λ ≥ 1, i = 1, 2.
In this paper, we discuss the G-decomposition of λKv into 6-circuits with two chords. We construct some holey G-designs using sharply 2-transitive group, and present the recursive structure by PBD. We also give a unified method to construct G-designs when the index equals the edge number of the discussed graph. Finally, the existence of G-GDλ（v） is given.
A directed triple system of order v, denoted by DTS(v, λ), is a pair (X,B) where X is a v-set and B is a collection of transitive triples on X such that every ordered pair of X belongs to λ triples of B. An overlarge set of disjoint DTS(v, λ), denoted by OLDTS(v, λ), is a collection {(Y\{y}, Ai)}i,such that Y is a (v + 1)-set, each (Y\{y}, Ai) is a DTS(v, λ) and all Ai's form a partition of all transitive triples of Y. In this paper, we shall discuss the existence problem of OLDTS(v, λ) and give the following conclusion: there exists an OLDTS(v, λ) if and only if either λ = 1 and v = 0, 1 (mod 3), or λ = 3 and v≠2.
Zi-hong TIAN~(1+) Li-jun JI~2 ~1 Institute of Mathematics,Hebei Normal University,Shijiazhuang 050016,China
A Mendelsohn (directed, or hybrid) triple system of order v, denoted by MTS(v,λ) (DTS(v,λ), or HTS(v,λ)), is a pair (X,B) where X is a v-set and B is a collection of some cyclic (transitive, or cyclic and transitive) triples on X such that every ordered pair of X belongs to λ triples of B. In this paper, a relation between three types of oriented triple systems was discussed. We conjecture: the block-incident graph of MTS(v,λ) is 3-edge colorable. Then we obtain three disjoint DTS(v,λ)s and four disjoint HTS(v,λ)s from a given MTS(v,λ).
Let λKv be the complete multigraph with v vertices and G 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, a simple graph G with 6 vertices and 7 edges is discussed, and the maximum G-PD(v) and the minimum G-CD(v) are constructed for all orders v.
