Packing bipartite graphs with covers of complete bipartite graphs

For a set S of graphs, a perfect S-packing (S-factor) of a graph G is a set of mutually vertex-disjoint subgraphs of G that each are isomorphic to a member of S and that together contain all vertices of G. If G allows a covering (locally bijective homomorphism) to a graph H, i.e., a vertex mapping  f:VG→VH satisfying the property that f(u)f(v) belongs to EH whenever the edge uv belongs to EG such that for every u∈VG the restriction of f to the neighborhood of u is bijective, then G is an H-cover. For some fixed H let S(H) consist of all connected H-covers. Let Kk,ℓ be the complete bipartite graph with partition classes of size k and ℓ, respectively. For all fixed k,ℓ≥1, we determine the computational complexity of the problem that tests whether a given bipartite graph has a perfect S(Kk,ℓ)-packing. Our technique is partially based on exploring a close relationship to pseudo-coverings. A pseudo-covering from a graph G to a graph H is a homomorphism from G to H that becomes a covering to H when restricted to a spanning subgraph of G. We settle the computational complexity of the problem that asks whether a graph allows a pseudo-covering to Kk,ℓ for all fixed k,ℓ≥1.