This work considers the cycle covering of complete graphs motivated by the design of survivable WDM networks, where the requests are routed on sub-networks which are protected independently from each other. The problem can be stated as follows: for a given graph $G$, find a cycle covering of the edge set of $K_n$, where $V(K_n) = V(G)$, such that each cycle in the covering satisfies the disjoint routing constraint (DRC), relatively to $G$, which can be stated as follows : to each edge of $K_n$ we associate in G a path and all the paths associated to the edges of a cycle of the covering must be vertex disjoint. Here we consider the case where $G = C_n$, a ring of size $n$ and we want to minimize the number of cycles in the covering. We give optimal solutions for the problem and as well as for variations of the problem, namely, its directed version and the case when cycle length is fixed as 4.