Orthogonal cycle systems with cycle length less than 10

Abstract

An H-decomposition of a graph 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 G is a set of kH-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 is an element of {5, 6, 7, 8, 9}, except when l = v.