\(C_{4}\)-Frames of \(M(b,n)\)

Let \( M(b, n) \) be the complete multipartite graph with \( b \) parts \( B_0, \ldots, B_{b-1} \) of size \( n \). A \( 4 \)-cycle system of \( M(b, n) \) is said to be a \emph{frame} if the \( 4 \)-cycles can be partitioned into sets \( S_1, \ldots, S_z \) such that for \( 1 \leq j \leq z \), \( S_j \) induces a \( 2 \)-factor of \( M(b, n) \setminus B_i \) for some \( i \in \mathbb{Z}_b \). The existence of a \( C_4 \)-frame of \( M(b, n) \) has been settled when \( n = 4 \) [6]. In this paper, we completely settle the existence question of a \( C_4 \)-frame of \( M(b, n) \) for all \( b \neq 2 \) and \( n \).