Orthogonal cycle systems with cycle length less than 10.

An H $H$‐decomposition of a graph G $G$ is a partition of the edge set of G $G$ into subsets, where each subset induces a copy of the graph H $H$. A k $k$‐orthogonal H $H$‐decomposition of G $G$ is a set of k $k$H $H$‐decompositions of G $G$ such that any two copies of H $H$ in distinct H $H$‐decompositions intersect in at most one edge. When G=Kv $G={K}_{v}$, we call the H $H$‐decomposition an H $H$‐system of order v $v$. In this paper, we consider the case H $H$ is an ℓ $\ell $‐cycle and construct a pair of orthogonal ℓ $\ell $‐cycle systems for all admissible orders when ℓ∈{5,6,7,8,9} $\ell \in \{5,6,7,8,9\}$, except when ℓ=v $\ell =v$. [ABSTRACT FROM AUTHOR]