α-resolvable Cycle Systems

In this paper, we discuss the existence of theα- resolvable cycle systems and frame generalizedbalanced tournament designs.LetλK_v(λDK_v) be (directed) complete multigraph of order v and indexλ. A cycle of length m is a sequence of m distinct vertices u₁,u₂,...,u_m, denoted by (u₁, u₂,..., u_m), and its edge set is {{u_i,u_i+1} : i = 1,2,...,m - 1}∪{{u₁,u_m}}. A directed cycle of length m is a sequence of m distinct vertices u₁,u₂,... ,u_m, denoted by 1,u₂,..., u_m>, and its directed edge set is {(u_i, u_i+1) : i = 1,2,..., m - 1}∪{(u_m, u₁)}. If the (directed) edges of aλK_v(λDK_v) can be decomposed into (directed) cycles of length m, then these (directed) cycles are called a (directed) m-cycle system, and denoted by m-CS(v,λ) (m-DCS(v,λ)). An m-CS(u,λ) (m-DCS(v,λ)) is said to beα-resolvable if its (directed) cycles can be partitioned into classes (calledα-resolution classes) such that each point of the design occurs in preciselyαcycles in each class. In the first part of this paper, we derive the existence of theα-resolvable 4-CS(v,λ) andα-resolvable m-DCS(v,λ) for m = 3,4.If the blocks of a (4,3)-GDD(X, (?),β) of type g^u can be arranged in to a (g^u/4)×gu array with properties: (1) the main diagonal consists of u empty subarrays of size (g/4)×g; (2) the blocks in each column form a partial 1-resolution classes partitioning X \G for some G∈(?), while the blocks in each row form a partial 4-resolution classes partitioning X\G for some G∈(?), then it is called a frame generalized balanced tournament design, and denoted by FGBTD(4, g^u). In the second part of this paper, we solve the existence of the FGBTD(4, g^u) basically.