Orthogonal cycle systems with cycle length less than 10

An ‐decomposition of a graph is a partition of the edge set of into subsets, where each subset induces a copy of the graph . A ‐orthogonal ‐decomposition of is a set of ‐decompositions of such that any two copies of in distinct ‐decompositions intersect in at most one edge. When , we call the ‐decomposition an ‐system of order . In this paper, we consider the case is an ‐cycle and construct a pair of orthogonal ‐cycle systems for all admissible orders when , except when .