On Cycle Frames with Cycles of Length \(8\)

Let \( M(b, n) \) be the complete multipartite graph with \( b \) parts \( B_0, \dots, B_{b-1} \) of size \( n \). A \( z \)-cycle system of \( M(b, n) \) is said to be a \emph{cycle-frame} if the \( z \)-cycles can be partitioned into sets \( S_1, \dots, S_k \) such that for \( 1 \leq j \leq k \), \( S_j \) induces a \( 2 \)-factor of \( M(b, n) \backslash B_i \) for some \( i \in \mathbb{Z}_b \). The existence of a \( C_z \)-frame of \( M(b, n) \) has been settled when \( z \in \{3, 4, 5, 6\} \). Here, we completely settle the case of \( C_z \)-frames when \( z \) is \( 8 \), and we give some solutions for larger values of \( z \).