Packing Bipartite Graphs with Covers of Complete Bipartite Graphs

Abstract

For a set $\mathcal{S}$ of graphs, a perfect $\mathcal{S}$-packing ($\mathcal{S}$-factor) of a graph G is a set of mutually vertex-disjoint subgraphs of G that each are isomorphic to a member of $\mathcal{S}$ 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 fixed H let $\mathcal{S}(H)$ consist of all H-covers. Let K _k,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 if a given bipartite graph has a perfect $\mathcal{S}(K_{k,\ell})$-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 if a graph allows a pseudo-covering to K _k,ℓ for all fixed k,ℓ ≥ 1.