Packing Bipartite Graphs with Covers of Complete Bipartite Graphs

摘要

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, then G is an H-cover. For some flxed H let S(H) consist of all H-covers. Let Kk,e be the complete bipartite graph with partition classes of size k and ￡, respectively. For all fixed k, e ≥ 1, we determine the computational complexity of the problem that tests if a given bipartite graph has a perfect S(Kk,e)-packmg. 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 if a graph allows a pseudo-covering to Kk,e for all fixed k,e≥l.