A maximum(v,G,λ)-PD and a minimum(v,G,λ)-CD are 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.
