Decomposition of complete bipartite multigraphs into paths and cycles having k edges

We give necessary and sufficient conditions for the decomposition of complete bipartite multigraph Km,n(λ) into paths and cycles having k edges. In particular, we show that such decomposition exists in Km,n(λ), when λ ≡ 0 (mod 2), m, n ≥ k2 , m + n > k, and k(p + q) = 2mn for k ≡ 0(mod 2) and also when λ ≥ 3, λm ≡ λn ≡ 0(mod 2), k(p + q) = λmn, m, n ≥ k, (resp., m,n ≥ 3k/2) for k ≡ 0(mod 4) (respectively, for k ≡ 2(mod 4)). In fact, the necessary conditions given above are also sufficient when λ = 2.