Multidecomposition of the complete graph into graph pairs of order 4 with various leaves

A graph-pair of order t is two non-isomorphic graphs G and H on t non-isolated vertices for which G ∪ H ∼= Kt for some integer t ≥ 4. Given a graph-pair (G,H), we say (G,H) divides some graph K if the edges of K can be partitioned into copies of G and H with at least one copy of G and at least one copy of H . We will refer to this partition as a (G,H)multidecomposition of K. In this paper, we consider the existence of multidecompositions of Kn−F for the graph-pair of order 4 where F is a Hamiltonian cycle, a 1-factor or almost 1-factor.