Decomposing the complete graph and the complete graph minus a 1-factor into copies of a graph G where G is the union of two disjoint cycles

Let G of order n be the vertex-disjoint union of two cycles. It is known that there exists a G-decomposition of Kv for all v≡1(mod2n). If G is bipartite and x is a positive integer, it is also known that there exists a G-decomposition of Knx−I, where I is a 1-factor. If G is not bipartite, there exists a G-decomposition of Kn if n is odd, and of Kn−I, where I is a 1-factor, if n is even. We use novel extensions of the Bose construction for Steiner triple systems and some recent results on the Oberwolfach Problem to obtain a G-decomposition of Kv for all v≡n(mod2n) when n is odd, unless G=C4∪C5 and v=9. If G consists of two odd cycles and n≡0(mod4), we also obtain a G-decomposition of Kv−I, for all v≡0(modn), v≠4n.