Packing bipartite graphs with covers of complete bipartite graphs

For a set of graphs, a perfect 4-packing (&-factor) of a graph G is a set of mutually vertex-disjoint subgraphs of G that each are isomorphic to a member of & 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 : V-G -> V-H satisfying the property that f (u)f (upsilon) belongs to EH whenever the edge u upsilon belongs to E-G such that for every u epsilon V-G the restriction off to the neighborhood of u is bijective, then G is an H-cover. For some fixed H let & (H) consist of all connected H-covers. Let K-k,K-l be the complete bipartite graph with partition classes of size k and E, 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 4(K-k,K-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 K(k,)l for all fixed k, l >= 1. (c) 2012 Elsevier B.V. All rights reserved.