Decomposition of complete multigraphs into crown graphs

Let λKv be the complete multigraph, G a finite simple graph. A G-design of λKv is denoted by GD(v,G,λ). The crown graph Qn is obtained by joining single pendant edge to each vertex of an n-cycle. We give new constructions for Qn-designs. Let v and λ be two positive integers. For n=4, 6, 8 and λ≥1, there exists a GD(v,Qn,λ) if and only if either (1) v>2n and λv(v−1)≡0 (mod 4n), or (2) v=2n and λ≡0 (mod 4). Let n≥4 be even. Then (1) there exists a GD(2n,Qn,λ) if and only if λ≡0 (mod 4). (2) There exists a GD(2n+1,Qn,λ) when λ≡0 (mod 4).