Orthogonal cycle systems with cycle length less than 10

Abstract

An $H$-decomposition of $G$ is a partition of the edge-set of $G$ into subsets, where each subset induces a copy of the graph $H$. A $k$-orthogonal $H$-decomposition of a graph $G$ is a set of $k$ $H$-decompositions of $G$, such that any two copies of $H$ in distinct $H$-decompositions intersect in at most one edge. When $G=K_v$ we call the $H$-decomposition an $H$-system of order $v$. In this paper we consider the case $H$ is an $l$-cycle and construct a pair of orthogonal $l$-cycle systems for all admissible orders when $l=5,6,7, 8\ or\ 9$, except $(l,v)=(7,7)$ and $(l,v)=(9,9)$.