Perfect Circle Quasi Bracket

Let (V,(?), B) be a (k, λ)-cycle group divisible design of type gu, in which the collection of blocks B can be partitioned into p (p≥0) parallel classes and du (d≥0) holey classes, each parallel class being a partition of V and each holey parallel class being a partition of V\Gi for some Gi∈(?),then (V,(?), B) is called a (k, λ)-cycle semiframe, denoted by (k, λ)-CSF(p, d, gu).If D=(V,(?),B) is a (k, λ)-CSF(p, d, gu). Suppose A(?)B satisfies the following conditions:A can be partitioned into u-1/u parallel classes;μA can be partitioned into u holey parallel classes with different holes, then A is called a μ-balanced set.Suppose D=(V,(?), B) is a (k,λ)-CSF(p, d, gu). If there are t mutually disjoint subsets A1,A2,...,At satisfying Ai (1≤i≤t) is a1-balanced set, then D is called a t-perfect cycle semiframe, denoted by (k, λ)-PCSFt(p, d, gu). Further, if B can be partitioned into t subsets A1,A2,...,At each Ai (1≤i≤t) is a1-balanced set, then the (k,λ)-PCSFt(p, d, gu) is called perfect cycle semiframe, denoted by (k,λ)-PCSF(gu).The existence of a (3,1)-PCSF(gu) has been almost solved recently, as well as some results on the existence of (3,λ)-PCSF(gu) with λ>1are obtained. In this paper, we nearly solve the existence of a (k,1)-PCSF(gu) for k∈{4,6}.