Packing Bipartite Graphs with Covers of Complete Bipartite Graphs

  • Jérémie Chalopin
  • Daniël Paulusma
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6078)


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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Angluin, D.: Local and global properties in networks of processors. In: 12th ACM Symposium on Theory of Computing, pp. 82–93. ACM, New York (1980)Google Scholar
  2. 2.
    Abello, J., Fellows, M.R., Stillwell, J.C.: On the complexity and combinatorics of covering finite complexes. Austral. J. Comb. 4, 103–112 (1991)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Biggs, N.: Constructing 5-arc transitive cubic graphs. J. London Math. Soc. II 26, 193–200 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chalopin, J.: Election and Local Computations on Closed Unlabelled Edges. In: Vojtáš, P., Bieliková, M., Charron-Bost, B., Sýkora, O. (eds.) SOFSEM 2005. LNCS, vol. 3381, pp. 81–90. Springer, Heidelberg (2005)Google Scholar
  5. 5.
    Chalopin, J., Paulusma, D.: Graph labelings derived from models in distributed computing. In: Fomin, F.V. (ed.) WG 2006. LNCS, vol. 4271, pp. 301–312. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  6. 6.
    Fiala, J.: Locally Injective Homomorphisms. Doctoral Thesis, Charles University (2000)Google Scholar
  7. 7.
    Fiala, J., Kratochvíl, J.: Locally constrained graph homomorphisms - structure, complexity, and applications. Comput. Sci. Rev. 2, 97–111 (2008)CrossRefzbMATHGoogle Scholar
  8. 8.
    Fiala, J., Paulusma, D.: A complete complexity classification of the role assignment problem. Theoret. Comput. Sci. 349, 67–81 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fiala, J., Paulusma, D., Telle, J.A.: Locally constrained graph homomorphism and equitable partitions. European J. Combin. 29, 850–880 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Garey, M.R., Johnson, D.R.: Computers and Intractability. Freeman, New York (1979)zbMATHGoogle Scholar
  11. 11.
    Hell, P.: Graph Packings. Elec. Notes Discrete Math. 5, 170–173 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hell, P., Kirkpatrick, D.G.: Scheduling, matching, and coloring. In: Alg. Methods in Graph Theory. Coll. Math. Soc. J. Bólyai 25, 273–279 (1981)Google Scholar
  13. 13.
    Hell, P., Kirkpatrick, D.G.: On the complexity of general graph factor problems. SIAM J. Comput. 12, 601–609 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hell, P., Kirkpatrick, D.G., Kratochvíl, J., Kříž, I.: On restricted two-factors. SIAM J. Discrete Math. 1, 472–484 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kratochvíl, J., Proskurowski, A., Telle, J.A.: Complexity of graph covering problem. Nordic. J. Comput. 5, 173–195 (1998)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Kratochvíl, J., Proskurowski, A., Telle, J.A.: Covering regular graphs. J. Combin. Theory Ser. B 71, 1–16 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Monnot, J., Toulouse, S.: The path partition problem and related problems in bipartite graphs. Oper. Res. Lett. 35, 677–684 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Plummer, M.D.: Graph factors and factorization: 1985-2003: A survey. Discrete Math. 307, 791–821 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Jérémie Chalopin
    • 1
  • Daniël Paulusma
    • 2
  1. 1.Laboratoire d’Informatique Fondamentale de MarseilleCNRS & Aix-Marseille UniversitéMarseilleFrance
  2. 2.Department of Computer ScienceDurham UniversityDurhamEngland

Personalised recommendations