Decomposition of bipartite graphs into spiders all of whose legs are two in length

As usual, K_m,n denotes the complete bipartite graph with parts of sizes m and n. For positive integers k ≤n, the crown C_n,k is the graph with vertex set {a₀, a₁,..., a_n-1, b₀, b₁,...,b_n-1} and edge set {a _ib_j: 0 ≤ i ≤ n - 1, j = i,i + 1,..., i + k - 1 (mod n)}. A spider is a tree with at most one vertex of degree more than two, called the center of the spider. A leg of a spider is a path from the center to a vertex of degree one. Let S_l(t) denote a spider of l legs, each of length t. An H-decomposition of a graph G is an edge-disjoint decomposition of G into copies of H. In this paper we investigate the problems of S _l(2)-decompositions of complete bipartite graphs and crowns, and prove that: (1) K_n,tl has an S_l(2)-decomposition if and only if nt ≡ 0 (mod 2), n≥ 2l if t = 1, and n ≥ J if t ≥ 2, (2) for t ≥ 2 and n ≥ tl, C_n,tl has an S_l(2)- decomposition if and only if nt ≡ 0 (mod 2), (3) for n ≥ 3t, C _n,3t has an S₃(2)-decomposition if and only if nt ≡ 0 (mod 2) and n ≡ 0 (mod 4) if t = 1.