Common Multiples of Paths and Stars with Crowns

A graph G is a common multiple of two graphs H₁ and H₂ if there exists a decomposition of G into edge-disjoint copies of H₁ and also a decomposition of G into edge-disjoint copies of H₂. If G is a common multiple of H₁ and H₂, and G has q edges, then we call G a (q, H₁, H₂) graph. Our paper deals with the following question: Given two graphs H₁ and H₂, for which values of q does there exist a (q, H₁, H₂) graph? when H₁ is either a path or a star with $3$ or $4$ edges and H₂ is a crown.