Isomorphic star decompositions of multicrowns and the power of cycles

For positive integers k ≤ n, the crown C_n,k is the graph with vertex set {a₁, a₂, ⋯, a_n, b₁, b₂, ⋯, b_n} and edge set {a_ib_j : 1 ≤ i ≤ n, j = i + 1, i + 2, ⋯, i + k (mod n)}. For any positive integer λ, the multicrown λC_n,k is the multiple graph obtained from the crown C_n,k by replacing each edge e by λ edges with the same end vertices as e. A star S_l is the complete bipartite graph K_1,l. If the edges of a graph G can be decomposed into subgraphs isomorphic to a graph H, then we say that G has an H-decomposition. In this paper, we prove that λC_n,k has an S_l-decomposition if and only if l ≤ k and λnk ≡ 0 (mod l). Thus, in particular, C_n,k has an S_l-decomposition if and only if l ≤ k and nk ≡ 0 (mod l). As a consequence, we show that if n ≥ 3, k < n/2, then C^k_n, the k-th power of the cycle C_n, has an S_l-decomposition if and only if l ≤ k + 1 and nk ≡ 0 (mod l).