Oriented Holey Frames With Block Size Three

An oriented holey frame is an oriented holey group divisible design whose blocks can be partitioned into several holey parallel classes,each of which is with respect to some group. Oriented holey frames can be used to construct some other designs such as oriented RGDD,oriented frames and so on.An incomplete oriented group divisible design is an oriented group divisible design satisfying every ordered pair of Y does not appears in any block,where Y is a subset of the vertex set X. It is powerful for the construction of other types of combinatorial designs.In this thesis,we investigate the existence of Mendelsohn(or Direcled)holey frame with block size 3 and incomplete Directed group divisible design with block size 4.We give the following results:There exists a(3,λ)-MHF(n,mt)(or(3,λ)-DHF(n,mt))if and only if λ≥1,n ≥ 4,t ≥3,(n-1)tm ≡0(mod 3),except possibly for A=1,(n,m,t)=(19,1,6),(n,m,t) ∈{(14,m,6),(15,m,6),(18,m,6),(23,m,6))with m≡1,5(mod 6).There exists a(4,λ)-IDGDD of type(g,h)u if and only if u ≥4,g ≥3h,λg(u-1)≡ 0(mod 3),λ(g-h)(u-1)≡0(mod 3),λu(u-1)(g2-h2)≡0(mod 6)with the possible exception of(λ,u,g,h)=(1,4,6,1).