Packing bipartite graphs with covers of complete bipartite graphs

  • Authors:

  • Jérémie Chalopin;Daniël Paulusma

  • Affiliations:

  • Laboratoire d'Informatique Fondamentale de Marseille, CNRS & Aix-Marseille Universitéé, Faculté des Sciences de Luminy, 13288 Marseille cedex 9, France;Department of Computer Science, Durham University, Science Laboratories, South Road, Durham DH1 3LE, England, United Kingdom

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2014

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Abstract

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:V"G-V"H satisfying the property that f(u)f(v) belongs to E"H whenever the edge uv belongs to E"G such that for every u@?V"G 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 K"k","@? be the complete bipartite graph with partition classes of size k and @?, respectively. For all fixed k,@?=1, we determine the computational complexity of the problem that tests whether a given bipartite graph has a perfect S(K"k","@?)-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","@? for all fixed k,@?=1.